| L(s) = 1 | − 0.446·2-s − 1.80·4-s + 2.33·5-s + 7-s + 1.69·8-s − 1.04·10-s − 4.53·11-s − 0.767·13-s − 0.446·14-s + 2.84·16-s − 3.43·17-s − 4.98·19-s − 4.20·20-s + 2.02·22-s + 4.14·23-s + 0.464·25-s + 0.342·26-s − 1.80·28-s + 9.37·29-s + 5.36·31-s − 4.66·32-s + 1.53·34-s + 2.33·35-s − 2.44·37-s + 2.22·38-s + 3.96·40-s + 1.04·41-s + ⋯ |
| L(s) = 1 | − 0.315·2-s − 0.900·4-s + 1.04·5-s + 0.377·7-s + 0.599·8-s − 0.329·10-s − 1.36·11-s − 0.212·13-s − 0.119·14-s + 0.711·16-s − 0.832·17-s − 1.14·19-s − 0.941·20-s + 0.431·22-s + 0.864·23-s + 0.0929·25-s + 0.0671·26-s − 0.340·28-s + 1.74·29-s + 0.964·31-s − 0.824·32-s + 0.262·34-s + 0.395·35-s − 0.402·37-s + 0.361·38-s + 0.627·40-s + 0.163·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.446T + 2T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 + 0.767T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.39T + 71T^{2} \) |
| 73 | \( 1 - 5.28T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−7.68040346805330134450282221335, −6.74692088109717100893133845563, −6.09646288993958756066335749990, −5.19938423816290574469223720996, −4.84086135079742606336177749368, −4.13172646152765065805132861528, −2.83853220459818778705997419510, −2.25235849150977401716595798371, −1.18771676966119885437651168629, 0,
1.18771676966119885437651168629, 2.25235849150977401716595798371, 2.83853220459818778705997419510, 4.13172646152765065805132861528, 4.84086135079742606336177749368, 5.19938423816290574469223720996, 6.09646288993958756066335749990, 6.74692088109717100893133845563, 7.68040346805330134450282221335
