| L(s) = 1 | + 0.0495·2-s − 3.19·3-s − 1.99·4-s + 5-s − 0.158·6-s − 0.197·8-s + 7.23·9-s + 0.0495·10-s − 11-s + 6.39·12-s + 2.44·13-s − 3.19·15-s + 3.98·16-s + 7.80·17-s + 0.358·18-s + 4.44·19-s − 1.99·20-s − 0.0495·22-s − 7.27·23-s + 0.633·24-s + 25-s + 0.120·26-s − 13.5·27-s − 9.41·29-s − 0.158·30-s − 10.0·31-s + 0.593·32-s + ⋯ |
| L(s) = 1 | + 0.0350·2-s − 1.84·3-s − 0.998·4-s + 0.447·5-s − 0.0646·6-s − 0.0699·8-s + 2.41·9-s + 0.0156·10-s − 0.301·11-s + 1.84·12-s + 0.676·13-s − 0.826·15-s + 0.996·16-s + 1.89·17-s + 0.0844·18-s + 1.01·19-s − 0.446·20-s − 0.0105·22-s − 1.51·23-s + 0.129·24-s + 0.200·25-s + 0.0237·26-s − 2.60·27-s − 1.74·29-s − 0.0289·30-s − 1.80·31-s + 0.104·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7517751063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7517751063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0495T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 - 0.538T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−9.148939479548905255465933622670, −7.76473234178677748822964044601, −7.43359559472242096830143124456, −6.05525539594655154682046980591, −5.61390611525718877770376297034, −5.35858421361251952079180425004, −4.21117976888340882858090458908, −3.54325158540255435120495844928, −1.59701351799255546659946374074, −0.63805526199710876782540217830,
0.63805526199710876782540217830, 1.59701351799255546659946374074, 3.54325158540255435120495844928, 4.21117976888340882858090458908, 5.35858421361251952079180425004, 5.61390611525718877770376297034, 6.05525539594655154682046980591, 7.43359559472242096830143124456, 7.76473234178677748822964044601, 9.148939479548905255465933622670
