L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 2·13-s + 14-s − 2·15-s + 16-s + 2·17-s + 18-s − 4·19-s − 2·20-s + 21-s − 22-s − 23-s + 24-s − 25-s − 2·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.353982756\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.353982756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.31128183729063, −15.85683627651636, −15.23695676557477, −14.87544217878818, −14.27819989446980, −13.86254771322700, −13.01904834697558, −12.54312137745790, −12.13381235976210, −11.30175185931273, −10.97535653297149, −10.15800700781161, −9.541212287178387, −8.724671114873346, −7.960813120428402, −7.753472672099828, −7.002693182070873, −6.282200645059709, −5.376263049563032, −4.812365642258237, −3.897827894082547, −3.722528610620907, −2.556834296214096, −2.078314476148422, −0.7267299120589836,
0.7267299120589836, 2.078314476148422, 2.556834296214096, 3.722528610620907, 3.897827894082547, 4.812365642258237, 5.376263049563032, 6.282200645059709, 7.002693182070873, 7.753472672099828, 7.960813120428402, 8.724671114873346, 9.541212287178387, 10.15800700781161, 10.97535653297149, 11.30175185931273, 12.13381235976210, 12.54312137745790, 13.01904834697558, 13.86254771322700, 14.27819989446980, 14.87544217878818, 15.23695676557477, 15.85683627651636, 16.31128183729063