| L(s) = 1 | + 2·2-s + 4·4-s + 19·5-s + 9·7-s + 8·8-s + 38·10-s + 13·11-s + 38·13-s + 18·14-s + 16·16-s − 99·17-s − 19·19-s + 76·20-s + 26·22-s − 68·23-s + 236·25-s + 76·26-s + 36·28-s − 130·29-s + 262·31-s + 32·32-s − 198·34-s + 171·35-s − 296·37-s − 38·38-s + 152·40-s + 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.69·5-s + 0.485·7-s + 0.353·8-s + 1.20·10-s + 0.356·11-s + 0.810·13-s + 0.343·14-s + 1/4·16-s − 1.41·17-s − 0.229·19-s + 0.849·20-s + 0.251·22-s − 0.616·23-s + 1.88·25-s + 0.573·26-s + 0.242·28-s − 0.832·29-s + 1.51·31-s + 0.176·32-s − 0.998·34-s + 0.825·35-s − 1.31·37-s − 0.162·38-s + 0.600·40-s + 0.0304·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.155864932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.155864932\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 5 | \( 1 - 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 13 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 99 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 - 262 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 8 T + p^{3} T^{2} \) |
| 43 | \( 1 - 73 T + p^{3} T^{2} \) |
| 47 | \( 1 - 271 T + p^{3} T^{2} \) |
| 53 | \( 1 - 502 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 587 T + p^{3} T^{2} \) |
| 67 | \( 1 - 684 T + p^{3} T^{2} \) |
| 71 | \( 1 + 992 T + p^{3} T^{2} \) |
| 73 | \( 1 + 507 T + p^{3} T^{2} \) |
| 79 | \( 1 - 980 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1046 T + p^{3} T^{2} \) |
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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
−11.05368055090480830879000824458, −10.33259624110105458003403724705, −9.281566863713514158321799515273, −8.407444470006520047535218802038, −6.82491974510227981146805197081, −6.13715598966095037397452676565, −5.25156137320193612903816016479, −4.10399019108168447441436297118, −2.47502364201397248140404160406, −1.53269995033226870726367889351,
1.53269995033226870726367889351, 2.47502364201397248140404160406, 4.10399019108168447441436297118, 5.25156137320193612903816016479, 6.13715598966095037397452676565, 6.82491974510227981146805197081, 8.407444470006520047535218802038, 9.281566863713514158321799515273, 10.33259624110105458003403724705, 11.05368055090480830879000824458
