| L(s) = 1 | − 0.506·2-s + 3-s − 1.74·4-s + 5-s − 0.506·6-s − 4.78·7-s + 1.89·8-s + 9-s − 0.506·10-s − 4.52·11-s − 1.74·12-s + 13-s + 2.42·14-s + 15-s + 2.52·16-s − 2.38·17-s − 0.506·18-s + 7.02·19-s − 1.74·20-s − 4.78·21-s + 2.29·22-s − 1.15·23-s + 1.89·24-s + 25-s − 0.506·26-s + 27-s + 8.34·28-s + ⋯ |
| L(s) = 1 | − 0.357·2-s + 0.577·3-s − 0.871·4-s + 0.447·5-s − 0.206·6-s − 1.80·7-s + 0.669·8-s + 0.333·9-s − 0.160·10-s − 1.36·11-s − 0.503·12-s + 0.277·13-s + 0.647·14-s + 0.258·15-s + 0.632·16-s − 0.577·17-s − 0.119·18-s + 1.61·19-s − 0.389·20-s − 1.04·21-s + 0.488·22-s − 0.241·23-s + 0.386·24-s + 0.200·25-s − 0.0992·26-s + 0.192·27-s + 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.506T + 2T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 7.02T + 19T^{2} \) |
| 23 | \( 1 + 1.15T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 6.86T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.76284656782083781338728994128, −7.18231506700174870984598359281, −6.31912505854070432840825423177, −5.58608778490472676109631236795, −4.84221525223192960492972083556, −3.91403469754262956911955770049, −3.05852445757666497038566976214, −2.61184603933533587132663069402, −1.10823242347320029807002105304, 0,
1.10823242347320029807002105304, 2.61184603933533587132663069402, 3.05852445757666497038566976214, 3.91403469754262956911955770049, 4.84221525223192960492972083556, 5.58608778490472676109631236795, 6.31912505854070432840825423177, 7.18231506700174870984598359281, 7.76284656782083781338728994128
