| L(s) = 1 | + 2-s − 0.0749·3-s + 4-s − 5-s − 0.0749·6-s + 7-s + 8-s − 2.99·9-s − 10-s − 0.301·11-s − 0.0749·12-s − 4.38·13-s + 14-s + 0.0749·15-s + 16-s + 4.30·17-s − 2.99·18-s − 7.08·19-s − 20-s − 0.0749·21-s − 0.301·22-s − 0.177·23-s − 0.0749·24-s + 25-s − 4.38·26-s + 0.449·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0432·3-s + 0.5·4-s − 0.447·5-s − 0.0306·6-s + 0.377·7-s + 0.353·8-s − 0.998·9-s − 0.316·10-s − 0.0910·11-s − 0.0216·12-s − 1.21·13-s + 0.267·14-s + 0.0193·15-s + 0.250·16-s + 1.04·17-s − 0.705·18-s − 1.62·19-s − 0.223·20-s − 0.0163·21-s − 0.0643·22-s − 0.0369·23-s − 0.0153·24-s + 0.200·25-s − 0.859·26-s + 0.0864·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345154640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345154640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0749T + 3T^{2} \) |
| 11 | \( 1 + 0.301T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 23 | \( 1 + 0.177T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 + 6.15T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.60T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
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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
−8.029178157399923691731151255888, −7.24303445128029514788431322741, −6.43755954758960894031529091749, −5.83147356239365498198183026504, −4.97101346645086717524751386849, −4.54226707734274954386060628563, −3.61477016533837964711497410633, −2.75755165930600631645285844877, −2.15718186015632359071739226224, −0.67972307577675164384469786224,
0.67972307577675164384469786224, 2.15718186015632359071739226224, 2.75755165930600631645285844877, 3.61477016533837964711497410633, 4.54226707734274954386060628563, 4.97101346645086717524751386849, 5.83147356239365498198183026504, 6.43755954758960894031529091749, 7.24303445128029514788431322741, 8.029178157399923691731151255888
