| L(s) = 1 | + 1.67·3-s − 3.48·5-s − 7-s − 0.193·9-s + 2.48·11-s + 13-s − 5.83·15-s + 5.02·17-s + 2.13·19-s − 1.67·21-s − 6.92·23-s + 7.11·25-s − 5.35·27-s + 0.612·29-s + 2.28·31-s + 4.15·33-s + 3.48·35-s − 8.09·37-s + 1.67·39-s − 6.31·41-s − 8.11·43-s + 0.675·45-s − 7.21·47-s + 49-s + 8.41·51-s + 12.6·53-s − 8.63·55-s + ⋯ |
| L(s) = 1 | + 0.967·3-s − 1.55·5-s − 0.377·7-s − 0.0646·9-s + 0.748·11-s + 0.277·13-s − 1.50·15-s + 1.21·17-s + 0.488·19-s − 0.365·21-s − 1.44·23-s + 1.42·25-s − 1.02·27-s + 0.113·29-s + 0.410·31-s + 0.723·33-s + 0.588·35-s − 1.33·37-s + 0.268·39-s − 0.985·41-s − 1.23·43-s + 0.100·45-s − 1.05·47-s + 0.142·49-s + 1.17·51-s + 1.74·53-s − 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 0.612T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 8.11T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 0.932T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 8.36T + 97T^{2} \) |
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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
−8.325477499025564382141663059820, −7.81667203913267345872150835996, −7.12865742016169870189413716170, −6.22950893870125877883512622429, −5.18974188754184674804491676630, −4.03158699384157137891715359924, −3.55943697210016536048377257086, −2.97578845893840946747609977342, −1.53377698942104893790111030014, 0,
1.53377698942104893790111030014, 2.97578845893840946747609977342, 3.55943697210016536048377257086, 4.03158699384157137891715359924, 5.18974188754184674804491676630, 6.22950893870125877883512622429, 7.12865742016169870189413716170, 7.81667203913267345872150835996, 8.325477499025564382141663059820
