| L(s) = 1 | + 2-s − 3-s + 4-s + 0.819·5-s − 6-s − 3.89·7-s + 8-s + 9-s + 0.819·10-s − 5.72·11-s − 12-s − 13-s − 3.89·14-s − 0.819·15-s + 16-s + 5.03·17-s + 18-s + 5.32·19-s + 0.819·20-s + 3.89·21-s − 5.72·22-s + 6.19·23-s − 24-s − 4.32·25-s − 26-s − 27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.366·5-s − 0.408·6-s − 1.47·7-s + 0.353·8-s + 0.333·9-s + 0.259·10-s − 1.72·11-s − 0.288·12-s − 0.277·13-s − 1.04·14-s − 0.211·15-s + 0.250·16-s + 1.22·17-s + 0.235·18-s + 1.22·19-s + 0.183·20-s + 0.849·21-s − 1.22·22-s + 1.29·23-s − 0.204·24-s − 0.865·25-s − 0.196·26-s − 0.192·27-s − 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 0.819T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 + 5.72T + 11T^{2} \) |
| 17 | \( 1 - 5.03T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 3.88T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 + 0.0410T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 2.29T + 61T^{2} \) |
| 67 | \( 1 + 1.59T + 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.310T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 0.414T + 97T^{2} \) |
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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
−7.24955008876500909795972282215, −6.78422701009062952797015623304, −5.75495271288448314656658154261, −5.54514687324608713164009427363, −4.93816576504824934154730748403, −3.83623575561588622652031160244, −2.99515740248746697949709939939, −2.66287937139959167494658779357, −1.21952918416460719160640352707, 0,
1.21952918416460719160640352707, 2.66287937139959167494658779357, 2.99515740248746697949709939939, 3.83623575561588622652031160244, 4.93816576504824934154730748403, 5.54514687324608713164009427363, 5.75495271288448314656658154261, 6.78422701009062952797015623304, 7.24955008876500909795972282215
