| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 6·11-s + 5·16-s − 4·18-s + 12·22-s + 6·23-s + 25-s − 12·29-s + 6·32-s − 6·36-s + 22·37-s − 20·43-s + 18·44-s + 12·46-s + 2·50-s + 6·53-s − 24·58-s + 7·64-s − 8·67-s + 24·71-s − 8·72-s + 44·74-s − 20·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2/3·9-s + 1.80·11-s + 5/4·16-s − 0.942·18-s + 2.55·22-s + 1.25·23-s + 1/5·25-s − 2.22·29-s + 1.06·32-s − 36-s + 3.61·37-s − 3.04·43-s + 2.71·44-s + 1.76·46-s + 0.282·50-s + 0.824·53-s − 3.15·58-s + 7/8·64-s − 0.977·67-s + 2.84·71-s − 0.942·72-s + 5.11·74-s − 2.25·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.705064387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.705064387\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.252183449285187563023939612845, −8.289976858527741605754067242124, −8.089955271560401233285711922708, −7.22555857938969291949122618245, −6.98245819905382774741247026763, −6.41172881162855055433374816405, −6.07165860333864637280028498246, −5.47184351442648669896536749746, −5.10121929575789116373472283151, −4.24154906072663690774936255935, −4.09357447562145058841583904603, −3.32460014230352801932133870733, −2.89619370582642367784100306287, −2.01395492425859231821166088910, −1.19460396262973182533759913525,
1.19460396262973182533759913525, 2.01395492425859231821166088910, 2.89619370582642367784100306287, 3.32460014230352801932133870733, 4.09357447562145058841583904603, 4.24154906072663690774936255935, 5.10121929575789116373472283151, 5.47184351442648669896536749746, 6.07165860333864637280028498246, 6.41172881162855055433374816405, 6.98245819905382774741247026763, 7.22555857938969291949122618245, 8.089955271560401233285711922708, 8.289976858527741605754067242124, 9.252183449285187563023939612845
