| L(s) = 1 | + 1.69·2-s − 3.03·3-s + 0.887·4-s − 2.24·5-s − 5.15·6-s + 4.06·7-s − 1.89·8-s + 6.19·9-s − 3.81·10-s − 2.69·12-s + 0.345·13-s + 6.91·14-s + 6.80·15-s − 4.98·16-s − 4.31·17-s + 10.5·18-s + 3.41·19-s − 1.99·20-s − 12.3·21-s + 5.21·23-s + 5.73·24-s + 0.0347·25-s + 0.586·26-s − 9.69·27-s + 3.60·28-s + 5.26·29-s + 11.5·30-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 1.75·3-s + 0.443·4-s − 1.00·5-s − 2.10·6-s + 1.53·7-s − 0.668·8-s + 2.06·9-s − 1.20·10-s − 0.776·12-s + 0.0957·13-s + 1.84·14-s + 1.75·15-s − 1.24·16-s − 1.04·17-s + 2.48·18-s + 0.783·19-s − 0.445·20-s − 2.69·21-s + 1.08·23-s + 1.17·24-s + 0.00695·25-s + 0.115·26-s − 1.86·27-s + 0.682·28-s + 0.977·29-s + 2.11·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.461165568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461165568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 13 | \( 1 - 0.345T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 0.807T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 6.80T + 59T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.61T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−7.41493321017697002130231738688, −7.18213577982047710924183785359, −6.19231232592012940538559563229, −5.58946808329497475515548749911, −4.97555504757358097685710929414, −4.49042629340684275818263163179, −4.12011965078542851644962293815, −2.99862985057671786580644088969, −1.65259655552709412718036580635, −0.57319472890414064261157859948,
0.57319472890414064261157859948, 1.65259655552709412718036580635, 2.99862985057671786580644088969, 4.12011965078542851644962293815, 4.49042629340684275818263163179, 4.97555504757358097685710929414, 5.58946808329497475515548749911, 6.19231232592012940538559563229, 7.18213577982047710924183785359, 7.41493321017697002130231738688
