| L(s) = 1 | − 4·2-s + 12·4-s + 10·5-s − 32·8-s + 8·9-s − 40·10-s − 26·13-s + 80·16-s − 32·18-s + 120·20-s + 75·25-s + 104·26-s − 64·29-s − 192·32-s + 96·36-s + 112·37-s − 320·40-s + 80·45-s + 98·49-s − 300·50-s − 312·52-s + 256·58-s + 224·61-s + 448·64-s − 260·65-s − 256·72-s − 32·73-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s + 8/9·9-s − 4·10-s − 2·13-s + 5·16-s − 1.77·18-s + 6·20-s + 3·25-s + 4·26-s − 2.20·29-s − 6·32-s + 8/3·36-s + 3.02·37-s − 8·40-s + 16/9·45-s + 2·49-s − 6·50-s − 6·52-s + 4.41·58-s + 3.67·61-s + 7·64-s − 4·65-s − 3.55·72-s − 0.438·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.241785302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241785302\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 8 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 488 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1048 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 1688 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1592 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6728 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 112 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8872 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
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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
−11.70704380890701659019018783284, −11.35556153998770912719557493259, −10.80255027991835859720054941993, −10.16946895793929817501194484759, −9.993158718705954345591794665537, −9.668619358144176472614824240176, −9.263893262317077804101630825631, −8.977455289239846522955268518162, −8.146704880545936852693593755381, −7.57622733341849399222363732473, −7.12480315982834030493386719687, −6.84404169312201391234964384349, −6.10241908711849516960472438900, −5.61809215852237298984843532045, −5.16543327595038031043707374389, −4.00191329161907589323504158474, −2.53816335291302214105978651693, −2.50622760018715897566178345924, −1.72177281088893856538788879985, −0.799416281286280986689910680605,
0.799416281286280986689910680605, 1.72177281088893856538788879985, 2.50622760018715897566178345924, 2.53816335291302214105978651693, 4.00191329161907589323504158474, 5.16543327595038031043707374389, 5.61809215852237298984843532045, 6.10241908711849516960472438900, 6.84404169312201391234964384349, 7.12480315982834030493386719687, 7.57622733341849399222363732473, 8.146704880545936852693593755381, 8.977455289239846522955268518162, 9.263893262317077804101630825631, 9.668619358144176472614824240176, 9.993158718705954345591794665537, 10.16946895793929817501194484759, 10.80255027991835859720054941993, 11.35556153998770912719557493259, 11.70704380890701659019018783284
