| L(s) = 1 | − 2.39·2-s + 3.00·3-s + 3.73·4-s − 1.01·5-s − 7.19·6-s − 4.72·7-s − 4.15·8-s + 6.02·9-s + 2.43·10-s + 0.589·11-s + 11.2·12-s − 6.17·13-s + 11.3·14-s − 3.05·15-s + 2.48·16-s + 2.09·17-s − 14.4·18-s + 19-s − 3.80·20-s − 14.2·21-s − 1.41·22-s + 5.81·23-s − 12.4·24-s − 3.96·25-s + 14.7·26-s + 9.09·27-s − 17.6·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.73·3-s + 1.86·4-s − 0.455·5-s − 2.93·6-s − 1.78·7-s − 1.47·8-s + 2.00·9-s + 0.771·10-s + 0.177·11-s + 3.24·12-s − 1.71·13-s + 3.02·14-s − 0.789·15-s + 0.621·16-s + 0.509·17-s − 3.40·18-s + 0.229·19-s − 0.850·20-s − 3.10·21-s − 0.300·22-s + 1.21·23-s − 2.55·24-s − 0.792·25-s + 2.89·26-s + 1.75·27-s − 3.33·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8690848959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8690848959\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.39T + 2T^{2} \) |
| 3 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 + 1.01T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 - 0.589T + 11T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−8.105267667704151281233475990355, −7.60882223216161675046920984845, −7.01757751151865125640352899338, −6.67544361135458528835717825236, −5.26421651299203523518478673118, −3.93804008196470079519251117750, −3.25711791492593292317907560061, −2.65571381974833850578526253324, −1.91424466480339590422744769547, −0.55088338904116448005782419965,
0.55088338904116448005782419965, 1.91424466480339590422744769547, 2.65571381974833850578526253324, 3.25711791492593292317907560061, 3.93804008196470079519251117750, 5.26421651299203523518478673118, 6.67544361135458528835717825236, 7.01757751151865125640352899338, 7.60882223216161675046920984845, 8.105267667704151281233475990355
