| L(s) = 1 | + 2.59·2-s − 2.99·3-s + 4.71·4-s − 7.75·6-s + 1.08·7-s + 7.02·8-s + 5.96·9-s + 11-s − 14.1·12-s + 3.56·13-s + 2.81·14-s + 8.76·16-s − 2.11·17-s + 15.4·18-s − 19-s − 3.25·21-s + 2.59·22-s + 4.81·23-s − 21.0·24-s + 9.23·26-s − 8.88·27-s + 5.11·28-s − 3.35·29-s + 4.72·31-s + 8.66·32-s − 2.99·33-s − 5.46·34-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 1.72·3-s + 2.35·4-s − 3.16·6-s + 0.410·7-s + 2.48·8-s + 1.98·9-s + 0.301·11-s − 4.07·12-s + 0.989·13-s + 0.751·14-s + 2.19·16-s − 0.512·17-s + 3.64·18-s − 0.229·19-s − 0.709·21-s + 0.552·22-s + 1.00·23-s − 4.29·24-s + 1.81·26-s − 1.70·27-s + 0.966·28-s − 0.622·29-s + 0.849·31-s + 1.53·32-s − 0.521·33-s − 0.938·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.267779472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.267779472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 + 0.737T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 9.96T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−7.76361465397836950867888115551, −6.84416476026100649844370003329, −6.54536027261380361087514595566, −5.79875517193705442800454262411, −5.36005168200134442503939957869, −4.55342826629033396377887672221, −4.17471381599132825282035207257, −3.18813299851725141964472235129, −1.96262923767217864928000990887, −0.977143253589717740937039333557,
0.977143253589717740937039333557, 1.96262923767217864928000990887, 3.18813299851725141964472235129, 4.17471381599132825282035207257, 4.55342826629033396377887672221, 5.36005168200134442503939957869, 5.79875517193705442800454262411, 6.54536027261380361087514595566, 6.84416476026100649844370003329, 7.76361465397836950867888115551
