L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 7-s + 3·8-s + 9-s − 6·11-s − 12-s − 2·13-s + 14-s − 16-s + 4·17-s − 18-s − 6·19-s − 21-s + 6·22-s + 3·24-s + 2·26-s + 27-s + 28-s − 2·29-s − 10·31-s − 5·32-s − 6·33-s − 4·34-s − 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s + 1.27·22-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.883·32-s − 1.04·33-s − 0.685·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.32936245971431996275671894173, −9.483915259822821092831866225661, −8.662795051661330682031069514757, −7.84120783606502230633349535272, −7.25875489058942760485085675987, −5.65015346036326179947249208182, −4.69294181049015431403759045817, −3.40857977083983573336135155062, −2.07938066921589939300536949382, 0,
2.07938066921589939300536949382, 3.40857977083983573336135155062, 4.69294181049015431403759045817, 5.65015346036326179947249208182, 7.25875489058942760485085675987, 7.84120783606502230633349535272, 8.662795051661330682031069514757, 9.483915259822821092831866225661, 10.32936245971431996275671894173