| L(s) = 1 | + 0.491·2-s − 2.84·3-s − 1.75·4-s − 1.40·6-s − 7-s − 1.84·8-s + 5.11·9-s + 3.49·11-s + 5.00·12-s + 0.0906·13-s − 0.491·14-s + 2.60·16-s − 2.50·17-s + 2.51·18-s − 3.09·19-s + 2.84·21-s + 1.71·22-s + 23-s + 5.26·24-s + 0.0445·26-s − 6.02·27-s + 1.75·28-s − 9.21·29-s + 7.09·31-s + 4.97·32-s − 9.94·33-s − 1.23·34-s + ⋯ |
| L(s) = 1 | + 0.347·2-s − 1.64·3-s − 0.879·4-s − 0.572·6-s − 0.377·7-s − 0.653·8-s + 1.70·9-s + 1.05·11-s + 1.44·12-s + 0.0251·13-s − 0.131·14-s + 0.651·16-s − 0.608·17-s + 0.593·18-s − 0.710·19-s + 0.621·21-s + 0.366·22-s + 0.208·23-s + 1.07·24-s + 0.00874·26-s − 1.15·27-s + 0.332·28-s − 1.71·29-s + 1.27·31-s + 0.880·32-s − 1.73·33-s − 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5532829003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5532829003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.491T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 0.0906T + 13T^{2} \) |
| 17 | \( 1 + 2.50T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 5.99T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 + 7.43T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 - 9.20T + 97T^{2} \) |
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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
−8.590179818108662884669411686304, −7.52217361131487746573005272829, −6.49173514228983186699715283646, −6.28997320243366809403754390786, −5.45253750994907193587455611087, −4.69822590083957559185164638284, −4.18242404733025471155728464779, −3.29380920925126133305539775642, −1.64572695335736017348186143471, −0.45276816533256622590101738162,
0.45276816533256622590101738162, 1.64572695335736017348186143471, 3.29380920925126133305539775642, 4.18242404733025471155728464779, 4.69822590083957559185164638284, 5.45253750994907193587455611087, 6.28997320243366809403754390786, 6.49173514228983186699715283646, 7.52217361131487746573005272829, 8.590179818108662884669411686304
