| L(s) = 1 | + 2.18·2-s + 3-s + 2.76·4-s + 5-s + 2.18·6-s − 1.86·7-s + 1.67·8-s + 9-s + 2.18·10-s − 6.19·11-s + 2.76·12-s + 3.05·13-s − 4.08·14-s + 15-s − 1.87·16-s − 2.25·17-s + 2.18·18-s − 1.15·19-s + 2.76·20-s − 1.86·21-s − 13.5·22-s + 1.67·24-s + 25-s + 6.66·26-s + 27-s − 5.17·28-s − 4.70·29-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.38·4-s + 0.447·5-s + 0.891·6-s − 0.706·7-s + 0.591·8-s + 0.333·9-s + 0.690·10-s − 1.86·11-s + 0.798·12-s + 0.846·13-s − 1.09·14-s + 0.258·15-s − 0.469·16-s − 0.546·17-s + 0.514·18-s − 0.265·19-s + 0.618·20-s − 0.407·21-s − 2.88·22-s + 0.341·24-s + 0.200·25-s + 1.30·26-s + 0.192·27-s − 0.977·28-s − 0.874·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.08930352890702618931770336500, −6.76151465376946255044926560818, −5.88128634216381346758165983674, −5.30432560686226935808670604342, −4.78952621846656629868687310007, −3.67744846987194292164862133619, −3.35813586654292297084568227614, −2.48432064751675757561379201132, −1.89391975192769330057344345534, 0,
1.89391975192769330057344345534, 2.48432064751675757561379201132, 3.35813586654292297084568227614, 3.67744846987194292164862133619, 4.78952621846656629868687310007, 5.30432560686226935808670604342, 5.88128634216381346758165983674, 6.76151465376946255044926560818, 7.08930352890702618931770336500
