| L(s) = 1 | − 2·5-s − 2·7-s + 6·11-s − 6·13-s − 10·17-s + 2·19-s + 2·23-s − 2·25-s − 8·29-s − 8·31-s + 4·35-s − 4·41-s − 8·43-s + 4·47-s + 3·49-s − 8·53-s − 12·55-s + 16·59-s + 8·61-s + 12·65-s + 6·67-s + 12·71-s − 12·77-s + 18·79-s − 4·83-s + 20·85-s + 4·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.80·11-s − 1.66·13-s − 2.42·17-s + 0.458·19-s + 0.417·23-s − 2/5·25-s − 1.48·29-s − 1.43·31-s + 0.676·35-s − 0.624·41-s − 1.21·43-s + 0.583·47-s + 3/7·49-s − 1.09·53-s − 1.61·55-s + 2.08·59-s + 1.02·61-s + 1.48·65-s + 0.733·67-s + 1.42·71-s − 1.36·77-s + 2.02·79-s − 0.439·83-s + 2.16·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7589773905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7589773905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 190 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69247814917445102402354737473, −7.55695829701518011535743341331, −6.95761701017399195808708798476, −6.84158340841337677529916553504, −6.69962296376422786067278444002, −6.42050509685313391210844701869, −5.76824538076009081581471509434, −5.44107036732067245304075133221, −4.92582765674797315601368394393, −4.91549410761598101225538044907, −4.08577905614069686340656597051, −4.07253397121681352589706421983, −3.59705079604356304886173667533, −3.56455118970851291484025486510, −2.73826941560852841838158480692, −2.42948670189715872435204141802, −1.84007508218933729871461112504, −1.74955773910235877915909269219, −0.70503064142347803198283574142, −0.26877759296438580179967345213,
0.26877759296438580179967345213, 0.70503064142347803198283574142, 1.74955773910235877915909269219, 1.84007508218933729871461112504, 2.42948670189715872435204141802, 2.73826941560852841838158480692, 3.56455118970851291484025486510, 3.59705079604356304886173667533, 4.07253397121681352589706421983, 4.08577905614069686340656597051, 4.91549410761598101225538044907, 4.92582765674797315601368394393, 5.44107036732067245304075133221, 5.76824538076009081581471509434, 6.42050509685313391210844701869, 6.69962296376422786067278444002, 6.84158340841337677529916553504, 6.95761701017399195808708798476, 7.55695829701518011535743341331, 7.69247814917445102402354737473
