| L(s) = 1 | + 3·3-s + 4·5-s + 7·7-s + 9·9-s + 26·11-s + 2·13-s + 12·15-s − 36·17-s + 76·19-s + 21·21-s + 114·23-s − 109·25-s + 27·27-s + 6·29-s + 256·31-s + 78·33-s + 28·35-s − 86·37-s + 6·39-s + 160·41-s + 220·43-s + 36·45-s − 308·47-s + 49·49-s − 108·51-s + 258·53-s + 104·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.357·5-s + 0.377·7-s + 1/3·9-s + 0.712·11-s + 0.0426·13-s + 0.206·15-s − 0.513·17-s + 0.917·19-s + 0.218·21-s + 1.03·23-s − 0.871·25-s + 0.192·27-s + 0.0384·29-s + 1.48·31-s + 0.411·33-s + 0.135·35-s − 0.382·37-s + 0.0246·39-s + 0.609·41-s + 0.780·43-s + 0.119·45-s − 0.955·47-s + 1/7·49-s − 0.296·51-s + 0.668·53-s + 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.729790786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.729790786\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 86 T + p^{3} T^{2} \) |
| 41 | \( 1 - 160 T + p^{3} T^{2} \) |
| 43 | \( 1 - 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 308 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 264 T + p^{3} T^{2} \) |
| 61 | \( 1 - 606 T + p^{3} T^{2} \) |
| 67 | \( 1 - 520 T + p^{3} T^{2} \) |
| 71 | \( 1 - 286 T + p^{3} T^{2} \) |
| 73 | \( 1 + 530 T + p^{3} T^{2} \) |
| 79 | \( 1 - 44 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 89 | \( 1 - 768 T + p^{3} T^{2} \) |
| 97 | \( 1 - 222 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18202526156211682633030459213, −10.02116976844018075672574879923, −9.256754511391465422248791912843, −8.392643222239171722915915647728, −7.35761502852889108004468996777, −6.34926502809716269383707534680, −5.08227801250171614477862726395, −3.91284713877592745281501384708, −2.59326374861362459413048405266, −1.22145569389880188051329271277,
1.22145569389880188051329271277, 2.59326374861362459413048405266, 3.91284713877592745281501384708, 5.08227801250171614477862726395, 6.34926502809716269383707534680, 7.35761502852889108004468996777, 8.392643222239171722915915647728, 9.256754511391465422248791912843, 10.02116976844018075672574879923, 11.18202526156211682633030459213
