| L(s) = 1 | − 5-s + 11-s + 13-s + 7·17-s − 19-s + 2·23-s + 25-s + 8·29-s − 2·31-s + 9·37-s − 41-s + 9·43-s − 7·47-s − 7·49-s + 12·53-s − 55-s − 10·59-s − 2·61-s − 65-s − 13·67-s − 6·71-s + 4·73-s + 2·79-s + 6·83-s − 7·85-s − 10·89-s + 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 0.277·13-s + 1.69·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s + 1.48·29-s − 0.359·31-s + 1.47·37-s − 0.156·41-s + 1.37·43-s − 1.02·47-s − 49-s + 1.64·53-s − 0.134·55-s − 1.30·59-s − 0.256·61-s − 0.124·65-s − 1.58·67-s − 0.712·71-s + 0.468·73-s + 0.225·79-s + 0.658·83-s − 0.759·85-s − 1.05·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.03456336923908, −13.50644957581961, −12.89813801768354, −12.50403348183630, −11.89141256257909, −11.75414066001107, −10.94846328342818, −10.61897141695689, −10.04558106950874, −9.516386195649734, −9.064610806336063, −8.408627227723854, −7.981519255257416, −7.531634683919459, −7.009142973613641, −6.265545082765259, −5.982702672140048, −5.250488929565690, −4.692794306246948, −4.143043111611197, −3.538218285055797, −2.970381669220342, −2.460466707836390, −1.295289944521438, −1.083777145550894, 0,
1.083777145550894, 1.295289944521438, 2.460466707836390, 2.970381669220342, 3.538218285055797, 4.143043111611197, 4.692794306246948, 5.250488929565690, 5.982702672140048, 6.265545082765259, 7.009142973613641, 7.531634683919459, 7.981519255257416, 8.408627227723854, 9.064610806336063, 9.516386195649734, 10.04558106950874, 10.61897141695689, 10.94846328342818, 11.75414066001107, 11.89141256257909, 12.50403348183630, 12.89813801768354, 13.50644957581961, 14.03456336923908
