| L(s) = 1 | − 1.66·2-s + 1.25·3-s + 0.769·4-s − 2.09·6-s + 4.13·7-s + 2.04·8-s − 1.41·9-s + 11-s + 0.969·12-s + 3.80·13-s − 6.88·14-s − 4.94·16-s − 2.64·17-s + 2.35·18-s + 19-s + 5.20·21-s − 1.66·22-s + 5.58·23-s + 2.57·24-s − 6.33·26-s − 5.55·27-s + 3.18·28-s − 6.34·29-s + 7.53·31-s + 4.13·32-s + 1.25·33-s + 4.39·34-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.726·3-s + 0.384·4-s − 0.855·6-s + 1.56·7-s + 0.723·8-s − 0.471·9-s + 0.301·11-s + 0.279·12-s + 1.05·13-s − 1.83·14-s − 1.23·16-s − 0.640·17-s + 0.555·18-s + 0.229·19-s + 1.13·21-s − 0.354·22-s + 1.16·23-s + 0.525·24-s − 1.24·26-s − 1.06·27-s + 0.601·28-s − 1.17·29-s + 1.35·31-s + 0.731·32-s + 0.219·33-s + 0.754·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680692090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680692090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 - 1.25T + 3T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−8.301193887977065025071357907046, −7.84113850641425422985983892311, −7.18734800208540035344673357397, −6.20365273459857074884090592319, −5.21630005487027909216909181086, −4.49453941692384015368445711721, −3.66804929827233011014680392521, −2.50625362706656426231885286789, −1.68859702361214157007263340083, −0.873705550631303583248271189942,
0.873705550631303583248271189942, 1.68859702361214157007263340083, 2.50625362706656426231885286789, 3.66804929827233011014680392521, 4.49453941692384015368445711721, 5.21630005487027909216909181086, 6.20365273459857074884090592319, 7.18734800208540035344673357397, 7.84113850641425422985983892311, 8.301193887977065025071357907046
