| L(s) = 1 | − 2.14·2-s + 3-s + 2.60·4-s + 5-s − 2.14·6-s − 2.89·7-s − 1.29·8-s + 9-s − 2.14·10-s − 2.09·11-s + 2.60·12-s − 1.63·13-s + 6.20·14-s + 15-s − 2.43·16-s + 3.96·17-s − 2.14·18-s + 3.33·19-s + 2.60·20-s − 2.89·21-s + 4.50·22-s − 6.77·23-s − 1.29·24-s + 25-s + 3.50·26-s + 27-s − 7.52·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.577·3-s + 1.30·4-s + 0.447·5-s − 0.875·6-s − 1.09·7-s − 0.456·8-s + 0.333·9-s − 0.678·10-s − 0.632·11-s + 0.751·12-s − 0.452·13-s + 1.65·14-s + 0.258·15-s − 0.608·16-s + 0.962·17-s − 0.505·18-s + 0.764·19-s + 0.581·20-s − 0.630·21-s + 0.960·22-s − 1.41·23-s − 0.263·24-s + 0.200·25-s + 0.687·26-s + 0.192·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 - 0.742T + 43T^{2} \) |
| 47 | \( 1 + 0.257T + 47T^{2} \) |
| 53 | \( 1 - 9.77T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + 0.741T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.76292425026094997247817991331, −7.42626455862296443978266249466, −6.52286048098264253206604958951, −5.87701150796166991209098533410, −4.90297183334532977807604902415, −3.75894014306190229581408427759, −2.87582261920515015824633440608, −2.21629830797975427694220270948, −1.15871797888256017314117787004, 0,
1.15871797888256017314117787004, 2.21629830797975427694220270948, 2.87582261920515015824633440608, 3.75894014306190229581408427759, 4.90297183334532977807604902415, 5.87701150796166991209098533410, 6.52286048098264253206604958951, 7.42626455862296443978266249466, 7.76292425026094997247817991331
