| L(s) = 1 | + 1.56·2-s + 0.438·4-s + 0.561·5-s − 7-s − 2.43·8-s + 0.876·10-s + 5.12·11-s + 0.561·13-s − 1.56·14-s − 4.68·16-s − 4·17-s − 4·19-s + 0.246·20-s + 8·22-s + 4.56·23-s − 4.68·25-s + 0.876·26-s − 0.438·28-s − 6.56·29-s + 6.56·31-s − 2.43·32-s − 6.24·34-s − 0.561·35-s − 4.56·37-s − 6.24·38-s − 1.36·40-s + 1.12·41-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s + 0.251·5-s − 0.377·7-s − 0.862·8-s + 0.277·10-s + 1.54·11-s + 0.155·13-s − 0.417·14-s − 1.17·16-s − 0.970·17-s − 0.917·19-s + 0.0550·20-s + 1.70·22-s + 0.951·23-s − 0.936·25-s + 0.171·26-s − 0.0828·28-s − 1.21·29-s + 1.17·31-s − 0.431·32-s − 1.07·34-s − 0.0949·35-s − 0.749·37-s − 1.01·38-s − 0.216·40-s + 0.175·41-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 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 + 6.56T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 3.68T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 8.87T + 97T^{2} \) |
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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
−7.08686145984257047277062581200, −6.62146242147116096697240737270, −6.03018424650913986328320914435, −5.44825375785914821239781166056, −4.32935013157507107495364988431, −4.18974682481240065892409091599, −3.30899433742353897524441807380, −2.46609911462089808027589725501, −1.45787859340109037237707780313, 0,
1.45787859340109037237707780313, 2.46609911462089808027589725501, 3.30899433742353897524441807380, 4.18974682481240065892409091599, 4.32935013157507107495364988431, 5.44825375785914821239781166056, 6.03018424650913986328320914435, 6.62146242147116096697240737270, 7.08686145984257047277062581200
