| L(s) = 1 | + 2.54·2-s − 1.35·3-s + 4.49·4-s − 3.45·6-s + 1.88·7-s + 6.34·8-s − 1.16·9-s + 0.101·11-s − 6.08·12-s + 2.74·13-s + 4.79·14-s + 7.19·16-s − 2.79·17-s − 2.96·18-s + 3.13·19-s − 2.55·21-s + 0.259·22-s + 5.40·23-s − 8.60·24-s + 7.00·26-s + 5.64·27-s + 8.45·28-s + 0.973·29-s − 7.26·31-s + 5.63·32-s − 0.138·33-s − 7.11·34-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.782·3-s + 2.24·4-s − 1.40·6-s + 0.711·7-s + 2.24·8-s − 0.387·9-s + 0.0307·11-s − 1.75·12-s + 0.762·13-s + 1.28·14-s + 1.79·16-s − 0.677·17-s − 0.698·18-s + 0.718·19-s − 0.556·21-s + 0.0553·22-s + 1.12·23-s − 1.75·24-s + 1.37·26-s + 1.08·27-s + 1.59·28-s + 0.180·29-s − 1.30·31-s + 0.995·32-s − 0.0240·33-s − 1.22·34-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
| \(L(1)\) |
\(\approx\) |
\(5.491507820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.491507820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 0.101T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 - 0.973T + 29T^{2} \) |
| 31 | \( 1 + 7.26T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 - 0.488T + 71T^{2} \) |
| 73 | \( 1 - 6.00T + 73T^{2} \) |
| 79 | \( 1 + 0.0506T + 79T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−7.73393484470149280646427669416, −7.03287997005498109026141443672, −6.32746134639907791727520562865, −5.74712631330099814633127125895, −5.18297215732996947840307884684, −4.62408368286137728558312245152, −3.83454860044774772107315312131, −3.02307141003887023466744598125, −2.16288368510272806565194972756, −1.01541306523134564639355665423,
1.01541306523134564639355665423, 2.16288368510272806565194972756, 3.02307141003887023466744598125, 3.83454860044774772107315312131, 4.62408368286137728558312245152, 5.18297215732996947840307884684, 5.74712631330099814633127125895, 6.32746134639907791727520562865, 7.03287997005498109026141443672, 7.73393484470149280646427669416
