| L(s) = 1 | + 3-s + 21.2·5-s + 31.6·7-s − 26·9-s − 0.712·11-s + 38.9·13-s + 21.2·15-s − 54.1·17-s − 19·19-s + 31.6·21-s − 61.7·23-s + 328.·25-s − 53·27-s + 225.·29-s + 131.·31-s − 0.712·33-s + 673.·35-s − 94.1·37-s + 38.9·39-s − 108.·41-s + 205.·43-s − 553.·45-s + 523.·47-s + 658.·49-s − 54.1·51-s − 560.·53-s − 15.1·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 1.90·5-s + 1.70·7-s − 0.962·9-s − 0.0195·11-s + 0.830·13-s + 0.366·15-s − 0.772·17-s − 0.229·19-s + 0.328·21-s − 0.560·23-s + 2.62·25-s − 0.377·27-s + 1.44·29-s + 0.762·31-s − 0.00375·33-s + 3.25·35-s − 0.418·37-s + 0.159·39-s − 0.414·41-s + 0.728·43-s − 1.83·45-s + 1.62·47-s + 1.91·49-s − 0.148·51-s − 1.45·53-s − 0.0371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.176469563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.176469563\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - T + 27T^{2} \) |
| 5 | \( 1 - 21.2T + 125T^{2} \) |
| 7 | \( 1 - 31.6T + 343T^{2} \) |
| 11 | \( 1 + 0.712T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 61.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 523.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 179.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 724.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 683.T + 9.12e5T^{2} \) |
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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
−9.143643637990003596728039989914, −8.658213100207168847929668342520, −7.997076187135639665498129466273, −6.63558425043175627857333775693, −5.92868618536553408998649041449, −5.23630052727194331490437794878, −4.39091155672215984628548418318, −2.74552881288828589826461210494, −2.01527396157207334617864057038, −1.13767762486003359729916879992,
1.13767762486003359729916879992, 2.01527396157207334617864057038, 2.74552881288828589826461210494, 4.39091155672215984628548418318, 5.23630052727194331490437794878, 5.92868618536553408998649041449, 6.63558425043175627857333775693, 7.997076187135639665498129466273, 8.658213100207168847929668342520, 9.143643637990003596728039989914
