| L(s) = 1 | − 1.16·2-s + 3-s − 0.634·4-s − 0.405·5-s − 1.16·6-s − 1.60·7-s + 3.07·8-s + 9-s + 0.473·10-s + 6.29·11-s − 0.634·12-s − 3.66·13-s + 1.88·14-s − 0.405·15-s − 2.32·16-s − 17-s − 1.16·18-s − 6.46·19-s + 0.257·20-s − 1.60·21-s − 7.35·22-s + 8.93·23-s + 3.07·24-s − 4.83·25-s + 4.28·26-s + 27-s + 1.02·28-s + ⋯ |
| L(s) = 1 | − 0.826·2-s + 0.577·3-s − 0.317·4-s − 0.181·5-s − 0.477·6-s − 0.608·7-s + 1.08·8-s + 0.333·9-s + 0.149·10-s + 1.89·11-s − 0.183·12-s − 1.01·13-s + 0.502·14-s − 0.104·15-s − 0.582·16-s − 0.242·17-s − 0.275·18-s − 1.48·19-s + 0.0575·20-s − 0.351·21-s − 1.56·22-s + 1.86·23-s + 0.628·24-s − 0.967·25-s + 0.840·26-s + 0.192·27-s + 0.192·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 5 | \( 1 + 0.405T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 0.701T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.81T + 59T^{2} \) |
| 61 | \( 1 - 0.764T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.249522989700331702175017829497, −7.41329907254372724553356289721, −6.88951551876076780605529199180, −6.10877069671935791316706460902, −4.83806094117153783308772955813, −4.15365105000470799643617117611, −3.50698299771869103209611653267, −2.26942612252022777779603077570, −1.30563726591846251460926617269, 0,
1.30563726591846251460926617269, 2.26942612252022777779603077570, 3.50698299771869103209611653267, 4.15365105000470799643617117611, 4.83806094117153783308772955813, 6.10877069671935791316706460902, 6.88951551876076780605529199180, 7.41329907254372724553356289721, 8.249522989700331702175017829497
