| L(s) = 1 | + 1.78·2-s − 1.18·3-s + 1.20·4-s − 0.957·5-s − 2.12·6-s + 4.30·7-s − 1.42·8-s − 1.59·9-s − 1.71·10-s − 5.53·11-s − 1.42·12-s + 2.24·13-s + 7.70·14-s + 1.13·15-s − 4.95·16-s − 5.07·17-s − 2.85·18-s − 19-s − 1.15·20-s − 5.09·21-s − 9.90·22-s + 5.97·23-s + 1.68·24-s − 4.08·25-s + 4.01·26-s + 5.44·27-s + 5.18·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.683·3-s + 0.601·4-s − 0.428·5-s − 0.865·6-s + 1.62·7-s − 0.503·8-s − 0.532·9-s − 0.541·10-s − 1.66·11-s − 0.411·12-s + 0.622·13-s + 2.05·14-s + 0.292·15-s − 1.23·16-s − 1.23·17-s − 0.673·18-s − 0.229·19-s − 0.257·20-s − 1.11·21-s − 2.11·22-s + 1.24·23-s + 0.344·24-s − 0.816·25-s + 0.787·26-s + 1.04·27-s + 0.979·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.193975520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193975520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 + 1.18T + 3T^{2} \) |
| 5 | \( 1 + 0.957T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 5.07T + 17T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 0.534T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 7.45T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 0.894T + 71T^{2} \) |
| 73 | \( 1 - 6.64T + 73T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
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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
−7.995474896050568820827722454271, −7.32196645279390252149246531117, −6.31440488383411436701069450040, −5.63024734990593984755272145548, −5.17875252526313751884598122773, −4.56571018933380420821812997741, −4.02205136151281998289317812846, −2.80471954042507583461321723341, −2.22798883158422759691903567411, −0.64114390274163272878182395634,
0.64114390274163272878182395634, 2.22798883158422759691903567411, 2.80471954042507583461321723341, 4.02205136151281998289317812846, 4.56571018933380420821812997741, 5.17875252526313751884598122773, 5.63024734990593984755272145548, 6.31440488383411436701069450040, 7.32196645279390252149246531117, 7.995474896050568820827722454271
