| L(s) = 1 | − 2·5-s + 7·7-s − 8·11-s − 42·13-s + 2·17-s + 124·19-s + 76·23-s − 121·25-s − 254·29-s + 72·31-s − 14·35-s + 398·37-s − 462·41-s − 212·43-s − 264·47-s + 49·49-s + 162·53-s + 16·55-s − 772·59-s + 30·61-s + 84·65-s + 764·67-s − 236·71-s + 418·73-s − 56·77-s − 552·79-s + 1.03e3·83-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 0.377·7-s − 0.219·11-s − 0.896·13-s + 0.0285·17-s + 1.49·19-s + 0.689·23-s − 0.967·25-s − 1.62·29-s + 0.417·31-s − 0.0676·35-s + 1.76·37-s − 1.75·41-s − 0.751·43-s − 0.819·47-s + 1/7·49-s + 0.419·53-s + 0.0392·55-s − 1.70·59-s + 0.0629·61-s + 0.160·65-s + 1.39·67-s − 0.394·71-s + 0.670·73-s − 0.0828·77-s − 0.786·79-s + 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 76 T + p^{3} T^{2} \) |
| 29 | \( 1 + 254 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 462 T + p^{3} T^{2} \) |
| 43 | \( 1 + 212 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 236 T + p^{3} T^{2} \) |
| 73 | \( 1 - 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1190 T + p^{3} T^{2} \) |
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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
−9.415517331800370074361017257866, −8.123144349559668497556454356195, −7.61812617425739965038183332281, −6.74274973600086939781891491422, −5.52384517615458771518220361493, −4.90860039948259659904186814860, −3.74475799209650779595340291297, −2.69360132997098156687949572082, −1.44699365226511919657586900368, 0,
1.44699365226511919657586900368, 2.69360132997098156687949572082, 3.74475799209650779595340291297, 4.90860039948259659904186814860, 5.52384517615458771518220361493, 6.74274973600086939781891491422, 7.61812617425739965038183332281, 8.123144349559668497556454356195, 9.415517331800370074361017257866
