| L(s) = 1 | + 2.32·2-s − 2.30·3-s + 3.41·4-s − 5.35·6-s + 3.59·7-s + 3.28·8-s + 2.30·9-s − 0.497·11-s − 7.85·12-s + 2.64·13-s + 8.36·14-s + 0.811·16-s + 5.10·17-s + 5.35·18-s − 0.987·19-s − 8.27·21-s − 1.15·22-s + 6.41·23-s − 7.55·24-s + 6.15·26-s + 1.61·27-s + 12.2·28-s − 5.57·29-s + 6.05·31-s − 4.67·32-s + 1.14·33-s + 11.8·34-s + ⋯ |
| L(s) = 1 | + 1.64·2-s − 1.32·3-s + 1.70·4-s − 2.18·6-s + 1.35·7-s + 1.16·8-s + 0.766·9-s − 0.150·11-s − 2.26·12-s + 0.734·13-s + 2.23·14-s + 0.202·16-s + 1.23·17-s + 1.26·18-s − 0.226·19-s − 1.80·21-s − 0.246·22-s + 1.33·23-s − 1.54·24-s + 1.20·26-s + 0.309·27-s + 2.31·28-s − 1.03·29-s + 1.08·31-s − 0.826·32-s + 0.199·33-s + 2.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.803459159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.803459159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 2.32T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 + 0.497T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 + 0.987T + 19T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 0.307T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 0.151T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 0.849T + 97T^{2} \) |
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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
−11.01257041233233443264048881131, −10.38501495678357883411442011263, −8.728480693427379835589229346589, −7.56471242597748593480285583346, −6.58160848217016626543322151190, −5.67035680537976072216711474466, −5.15575105085395117118524440567, −4.43066785138020503205035849595, −3.18242993964173064081435791134, −1.46769623369423430226905813363,
1.46769623369423430226905813363, 3.18242993964173064081435791134, 4.43066785138020503205035849595, 5.15575105085395117118524440567, 5.67035680537976072216711474466, 6.58160848217016626543322151190, 7.56471242597748593480285583346, 8.728480693427379835589229346589, 10.38501495678357883411442011263, 11.01257041233233443264048881131
