| L(s) = 1 | − 5-s + 5.70·11-s + 6.70·13-s − 3.44·17-s − 1.25·19-s − 7.96·23-s + 25-s − 7.18·29-s + 31-s − 8.96·37-s − 10.2·41-s − 4.44·43-s − 1.48·47-s + 6.89·53-s − 5.70·55-s + 0.293·59-s + 0.259·61-s − 6.70·65-s − 6.70·67-s − 11.7·71-s − 0.966·73-s − 11.6·79-s + 14.8·83-s + 3.44·85-s + 12.6·89-s + 1.25·95-s + 13.1·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.72·11-s + 1.86·13-s − 0.836·17-s − 0.288·19-s − 1.66·23-s + 0.200·25-s − 1.33·29-s + 0.179·31-s − 1.47·37-s − 1.59·41-s − 0.678·43-s − 0.216·47-s + 0.947·53-s − 0.769·55-s + 0.0381·59-s + 0.0332·61-s − 0.831·65-s − 0.819·67-s − 1.38·71-s − 0.113·73-s − 1.30·79-s + 1.63·83-s + 0.373·85-s + 1.33·89-s + 0.129·95-s + 1.33·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5.70T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 + 7.96T + 23T^{2} \) |
| 29 | \( 1 + 7.18T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 8.96T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 - 0.293T + 59T^{2} \) |
| 61 | \( 1 - 0.259T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 0.966T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 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
−7.36825270394337514533168206342, −6.50471220730252947194194981301, −6.31659063927488046482924852093, −5.42449631077325060868522431512, −4.35028462576850072862466286940, −3.79477823649739910477239591833, −3.43822380176136646210394152842, −1.93584355132356114306191711914, −1.38362959492251503507087509907, 0,
1.38362959492251503507087509907, 1.93584355132356114306191711914, 3.43822380176136646210394152842, 3.79477823649739910477239591833, 4.35028462576850072862466286940, 5.42449631077325060868522431512, 6.31659063927488046482924852093, 6.50471220730252947194194981301, 7.36825270394337514533168206342
