| L(s) = 1 | − 1.70·3-s + 1.80·5-s − 0.510·7-s − 0.0863·9-s + 2.94·11-s − 2.13·13-s − 3.07·15-s − 5.07·17-s − 3.44·19-s + 0.870·21-s − 5.81·23-s − 1.74·25-s + 5.26·27-s − 3.06·29-s + 0.190·31-s − 5.02·33-s − 0.919·35-s − 2.93·37-s + 3.64·39-s − 5.04·41-s + 4.54·43-s − 0.155·45-s + 3.80·47-s − 6.73·49-s + 8.65·51-s + 12.5·53-s + 5.30·55-s + ⋯ |
| L(s) = 1 | − 0.985·3-s + 0.806·5-s − 0.192·7-s − 0.0287·9-s + 0.887·11-s − 0.591·13-s − 0.794·15-s − 1.22·17-s − 0.789·19-s + 0.190·21-s − 1.21·23-s − 0.349·25-s + 1.01·27-s − 0.568·29-s + 0.0341·31-s − 0.874·33-s − 0.155·35-s − 0.482·37-s + 0.583·39-s − 0.788·41-s + 0.693·43-s − 0.0232·45-s + 0.554·47-s − 0.962·49-s + 1.21·51-s + 1.71·53-s + 0.715·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9660539540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9660539540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 0.510T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 5.07T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 5.81T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 - 0.190T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 + 6.72T + 61T^{2} \) |
| 67 | \( 1 + 1.34T + 67T^{2} \) |
| 71 | \( 1 - 5.95T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−7.75745113454287357749918750506, −6.73814702840173833783634138446, −6.47215434440604945895635835197, −5.81221719741764527889933727237, −5.20210351395265289980327327087, −4.38140971107074023724496187599, −3.68620594220023800575826565720, −2.38615611797700564112297530164, −1.85746384740789531480848319787, −0.49205764113265197160013529979,
0.49205764113265197160013529979, 1.85746384740789531480848319787, 2.38615611797700564112297530164, 3.68620594220023800575826565720, 4.38140971107074023724496187599, 5.20210351395265289980327327087, 5.81221719741764527889933727237, 6.47215434440604945895635835197, 6.73814702840173833783634138446, 7.75745113454287357749918750506
