| L(s) = 1 | − 2·2-s + 2·3-s − 4-s − 4·6-s − 3·7-s + 5·8-s − 4·9-s + 8·11-s − 2·12-s + 6·14-s − 16-s + 2·17-s + 8·18-s + 4·19-s − 6·21-s − 16·22-s + 5·23-s + 10·24-s − 13·27-s + 3·28-s − 29-s + 5·31-s − 4·32-s + 16·33-s − 4·34-s + 4·36-s + 12·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s − 1.13·7-s + 1.76·8-s − 4/3·9-s + 2.41·11-s − 0.577·12-s + 1.60·14-s − 1/4·16-s + 0.485·17-s + 1.88·18-s + 0.917·19-s − 1.30·21-s − 3.41·22-s + 1.04·23-s + 2.04·24-s − 2.50·27-s + 0.566·28-s − 0.185·29-s + 0.898·31-s − 0.707·32-s + 2.78·33-s − 0.685·34-s + 2/3·36-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212774877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212774877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 17 T^{2} + 29 T^{3} + 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 8 T + 52 T^{2} - 189 T^{3} + 52 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 36 T^{2} - 81 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 46 T^{2} - 151 T^{3} + 46 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 5 T + 68 T^{2} - 217 T^{3} + 68 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 43 T^{2} + 141 T^{3} + 43 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 57 T^{2} - 143 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 12 T + 152 T^{2} - 917 T^{3} + 152 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 7 T + 74 T^{2} - 623 T^{3} + 74 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 13 T + 169 T^{2} + 1105 T^{3} + 169 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 18 T + 242 T^{2} + 1859 T^{3} + 242 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + T + 73 T^{2} - 231 T^{3} + 73 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 19 T + 260 T^{2} - 2243 T^{3} + 260 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 4 T + 116 T^{2} - 249 T^{3} + 116 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + T + 129 T^{2} + 175 T^{3} + 129 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 27 T + 435 T^{2} - 4381 T^{3} + 435 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 403 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 5 T + 75 T^{2} + 917 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 109 T^{2} + 1365 T^{3} + 109 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 11 T + 193 T^{2} - 1677 T^{3} + 193 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 7 T + 207 T^{2} - 1057 T^{3} + 207 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78282134278137484054348086988, −7.30762956595311795838772429266, −6.85313344313831315091575158674, −6.70435445825593970846163217796, −6.48548560202082612338444332159, −6.36457701048076730794612420945, −6.24378716529769598166014586979, −5.57242289370117228857805972082, −5.40614890176941124655801011911, −5.34458659830089331886437301952, −4.78896980222535454526839708947, −4.75009897893513764672079986238, −4.19604884041521069167132729151, −3.93453060831937861384247281224, −3.72287987979629760065920663063, −3.48701411246965342406723250802, −3.13395596815571545867208734597, −3.04084159320553945452029450296, −2.68910148034180893436344170914, −2.38608790941175170417396905584, −1.73341428399287427319682538398, −1.56299047990019636835082121157, −0.861773826435652521127232430657, −0.837428818704594714155957021262, −0.35557803849242950556701049601,
0.35557803849242950556701049601, 0.837428818704594714155957021262, 0.861773826435652521127232430657, 1.56299047990019636835082121157, 1.73341428399287427319682538398, 2.38608790941175170417396905584, 2.68910148034180893436344170914, 3.04084159320553945452029450296, 3.13395596815571545867208734597, 3.48701411246965342406723250802, 3.72287987979629760065920663063, 3.93453060831937861384247281224, 4.19604884041521069167132729151, 4.75009897893513764672079986238, 4.78896980222535454526839708947, 5.34458659830089331886437301952, 5.40614890176941124655801011911, 5.57242289370117228857805972082, 6.24378716529769598166014586979, 6.36457701048076730794612420945, 6.48548560202082612338444332159, 6.70435445825593970846163217796, 6.85313344313831315091575158674, 7.30762956595311795838772429266, 7.78282134278137484054348086988
