| L(s) = 1 | − 3.30·3-s − 2.23·5-s + 4.70·7-s + 7.91·9-s + 3.99·11-s − 1.48·13-s + 7.39·15-s + 7.43·17-s − 5.93·19-s − 15.5·21-s − 0.000615·23-s + 0.0140·25-s − 16.2·27-s − 9.83·29-s − 5.34·31-s − 13.2·33-s − 10.5·35-s + 8.90·37-s + 4.91·39-s − 4.14·41-s + 6.71·43-s − 17.7·45-s − 1.65·47-s + 15.1·49-s − 24.5·51-s − 6.85·53-s − 8.95·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s − 1.00·5-s + 1.77·7-s + 2.63·9-s + 1.20·11-s − 0.412·13-s + 1.90·15-s + 1.80·17-s − 1.36·19-s − 3.39·21-s − 0.000128·23-s + 0.00281·25-s − 3.12·27-s − 1.82·29-s − 0.960·31-s − 2.29·33-s − 1.78·35-s + 1.46·37-s + 0.787·39-s − 0.647·41-s + 1.02·43-s − 2.64·45-s − 0.240·47-s + 2.16·49-s − 3.43·51-s − 0.941·53-s − 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 0.000615T + 23T^{2} \) |
| 29 | \( 1 + 9.83T + 29T^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 6.71T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.62T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 0.282T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.51550637269749913156795046170, −7.19534540712987556244145105346, −6.01195808321742562430846098894, −5.69347059866342708178384291990, −4.75977852847612848451736513532, −4.32440931200672498417533910765, −3.70682795816722086792172943209, −1.78518473071641640697545349027, −1.19664573267118101904774789504, 0,
1.19664573267118101904774789504, 1.78518473071641640697545349027, 3.70682795816722086792172943209, 4.32440931200672498417533910765, 4.75977852847612848451736513532, 5.69347059866342708178384291990, 6.01195808321742562430846098894, 7.19534540712987556244145105346, 7.51550637269749913156795046170
