| L(s) = 1 | + 2.51·2-s + 0.981·3-s + 4.34·4-s + 2.47·6-s − 3.43·7-s + 5.91·8-s − 2.03·9-s − 1.87·11-s + 4.26·12-s − 1.10·13-s − 8.64·14-s + 6.21·16-s − 7.45·17-s − 5.13·18-s + 0.382·19-s − 3.36·21-s − 4.71·22-s + 0.968·23-s + 5.81·24-s − 2.79·26-s − 4.94·27-s − 14.9·28-s − 1.15·29-s + 7.77·31-s + 3.82·32-s − 1.83·33-s − 18.7·34-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.566·3-s + 2.17·4-s + 1.00·6-s − 1.29·7-s + 2.09·8-s − 0.678·9-s − 0.564·11-s + 1.23·12-s − 0.307·13-s − 2.31·14-s + 1.55·16-s − 1.80·17-s − 1.20·18-s + 0.0877·19-s − 0.734·21-s − 1.00·22-s + 0.202·23-s + 1.18·24-s − 0.548·26-s − 0.951·27-s − 2.81·28-s − 0.214·29-s + 1.39·31-s + 0.676·32-s − 0.319·33-s − 3.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 - 0.981T + 3T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 19 | \( 1 - 0.382T + 19T^{2} \) |
| 23 | \( 1 - 0.968T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 + 0.798T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 + 6.40T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 6.79T + 83T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−7.40350492692845026143242272024, −6.68161908036408636634995845891, −6.26789580383756681448264736798, −5.50678306235465618986739597428, −4.72899160272463949721918750808, −4.04076565404306033733337642149, −3.14354886277671518910407259874, −2.78719987469887356511762946331, −2.04976697619626294716379407003, 0,
2.04976697619626294716379407003, 2.78719987469887356511762946331, 3.14354886277671518910407259874, 4.04076565404306033733337642149, 4.72899160272463949721918750808, 5.50678306235465618986739597428, 6.26789580383756681448264736798, 6.68161908036408636634995845891, 7.40350492692845026143242272024
