| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 4.49·7-s + 2.23·8-s + 2.54·11-s − 1.57·13-s − 2.77·14-s + 1.85·16-s + 2.46·17-s + 3.04·19-s − 1.57·22-s + 7.23·23-s + 0.971·26-s − 7.27·28-s + 4.21·29-s + 4.39·31-s − 5.61·32-s − 1.52·34-s + 37-s − 1.87·38-s + 4.42·41-s − 0.207·43-s − 4.11·44-s − 4.46·46-s + 0.746·47-s + 13.2·49-s + 2.54·52-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 1.69·7-s + 0.790·8-s + 0.766·11-s − 0.435·13-s − 0.742·14-s + 0.463·16-s + 0.598·17-s + 0.697·19-s − 0.335·22-s + 1.50·23-s + 0.190·26-s − 1.37·28-s + 0.782·29-s + 0.789·31-s − 0.993·32-s − 0.261·34-s + 0.164·37-s − 0.304·38-s + 0.691·41-s − 0.0316·43-s − 0.620·44-s − 0.658·46-s + 0.108·47-s + 1.88·49-s + 0.352·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.127919353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127919353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 4.49T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 + 0.207T + 43T^{2} \) |
| 47 | \( 1 - 0.746T + 47T^{2} \) |
| 53 | \( 1 + 8.08T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 + 3.31T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 1.72T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.88670178483331505005958900052, −7.39924594792951335818369469305, −6.55314244109911223385024642941, −5.43064362632840141814807990078, −4.98000618197452913405726960233, −4.45162915792126516839404666547, −3.62489778798703115931762979227, −2.54722546112199859585829423511, −1.35311400056396735153413991683, −0.935413660374392224977203634106,
0.935413660374392224977203634106, 1.35311400056396735153413991683, 2.54722546112199859585829423511, 3.62489778798703115931762979227, 4.45162915792126516839404666547, 4.98000618197452913405726960233, 5.43064362632840141814807990078, 6.55314244109911223385024642941, 7.39924594792951335818369469305, 7.88670178483331505005958900052
