| L(s) = 1 | − 0.311·2-s − 2.90·3-s − 1.90·4-s + 0.903·6-s + 0.903·7-s + 1.21·8-s + 5.42·9-s − 1.52·11-s + 5.52·12-s + 0.622·13-s − 0.280·14-s + 3.42·16-s + 7.95·17-s − 1.68·18-s − 1.09·19-s − 2.62·21-s + 0.474·22-s − 7.52·23-s − 3.52·24-s − 0.193·26-s − 7.05·27-s − 1.71·28-s − 29-s − 6.90·31-s − 3.49·32-s + 4.42·33-s − 2.47·34-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 1.67·3-s − 0.951·4-s + 0.368·6-s + 0.341·7-s + 0.429·8-s + 1.80·9-s − 0.459·11-s + 1.59·12-s + 0.172·13-s − 0.0750·14-s + 0.857·16-s + 1.92·17-s − 0.398·18-s − 0.251·19-s − 0.572·21-s + 0.101·22-s − 1.56·23-s − 0.719·24-s − 0.0379·26-s − 1.35·27-s − 0.324·28-s − 0.185·29-s − 1.23·31-s − 0.617·32-s + 0.770·33-s − 0.424·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−10.14947908270234397163706403035, −9.376248230669232479291433484598, −8.072204455606458416156512160352, −7.47364664178704288701865864094, −6.05143608826649135083135865839, −5.51789525488542379787768533928, −4.72223352099411398645381805121, −3.69515588520996920983369015684, −1.39422349284949440368542575584, 0,
1.39422349284949440368542575584, 3.69515588520996920983369015684, 4.72223352099411398645381805121, 5.51789525488542379787768533928, 6.05143608826649135083135865839, 7.47364664178704288701865864094, 8.072204455606458416156512160352, 9.376248230669232479291433484598, 10.14947908270234397163706403035
