| L(s) = 1 | + 2-s + 3.17·3-s + 4-s + 2.00·5-s + 3.17·6-s − 7-s + 8-s + 7.06·9-s + 2.00·10-s + 1.19·11-s + 3.17·12-s − 5.69·13-s − 14-s + 6.37·15-s + 16-s + 5.94·17-s + 7.06·18-s + 5.24·19-s + 2.00·20-s − 3.17·21-s + 1.19·22-s + 5.18·23-s + 3.17·24-s − 0.964·25-s − 5.69·26-s + 12.8·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.83·3-s + 0.5·4-s + 0.898·5-s + 1.29·6-s − 0.377·7-s + 0.353·8-s + 2.35·9-s + 0.635·10-s + 0.361·11-s + 0.915·12-s − 1.58·13-s − 0.267·14-s + 1.64·15-s + 0.250·16-s + 1.44·17-s + 1.66·18-s + 1.20·19-s + 0.449·20-s − 0.692·21-s + 0.255·22-s + 1.08·23-s + 0.647·24-s − 0.192·25-s − 1.11·26-s + 2.47·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.679419798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.679419798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 4.15T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 0.244T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 9.05T + 83T^{2} \) |
| 89 | \( 1 + 3.85T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.70554784997927893144642194417, −7.53311534057781751712735663116, −6.92098492140780072577007763998, −5.72360225485862981487636517689, −5.24714924382012551575366292945, −4.21778733967178211155234040840, −3.39643224218116699515920705241, −2.92816947423295019248580082106, −2.13899512680586136120825997405, −1.38231450611053164029042767137,
1.38231450611053164029042767137, 2.13899512680586136120825997405, 2.92816947423295019248580082106, 3.39643224218116699515920705241, 4.21778733967178211155234040840, 5.24714924382012551575366292945, 5.72360225485862981487636517689, 6.92098492140780072577007763998, 7.53311534057781751712735663116, 7.70554784997927893144642194417
