| L(s) = 1 | − 2.22·2-s + 1.03·3-s + 2.94·4-s − 2.30·6-s + 2.02·7-s − 2.09·8-s − 1.92·9-s + 0.0848·11-s + 3.04·12-s + 5.72·13-s − 4.50·14-s − 1.23·16-s + 2.53·17-s + 4.27·18-s + 2.10·21-s − 0.188·22-s + 0.309·23-s − 2.16·24-s − 12.7·26-s − 5.10·27-s + 5.95·28-s − 2.62·29-s − 8.07·31-s + 6.92·32-s + 0.0879·33-s − 5.62·34-s − 5.65·36-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.598·3-s + 1.47·4-s − 0.941·6-s + 0.765·7-s − 0.739·8-s − 0.641·9-s + 0.0255·11-s + 0.880·12-s + 1.58·13-s − 1.20·14-s − 0.308·16-s + 0.613·17-s + 1.00·18-s + 0.458·21-s − 0.0401·22-s + 0.0644·23-s − 0.442·24-s − 2.49·26-s − 0.982·27-s + 1.12·28-s − 0.487·29-s − 1.44·31-s + 1.22·32-s + 0.0153·33-s − 0.964·34-s − 0.943·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253109258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253109258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 1.03T + 3T^{2} \) |
| 7 | \( 1 - 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.0848T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 23 | \( 1 - 0.309T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 + 0.650T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 14.5T + 53T^{2} \) |
| 59 | \( 1 + 7.47T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 0.115T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 0.225T + 97T^{2} \) |
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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
−7.938228196652915105973268773811, −7.45738154251477171724410622108, −6.63909590070426471146028025755, −5.81907521434106327455893405804, −5.15338505540566824716460038141, −3.92490546540567348099013013340, −3.33582047119287687103450576286, −2.22213837184339648666915991964, −1.62238806367056848927661365934, −0.69202828714867567264399663876,
0.69202828714867567264399663876, 1.62238806367056848927661365934, 2.22213837184339648666915991964, 3.33582047119287687103450576286, 3.92490546540567348099013013340, 5.15338505540566824716460038141, 5.81907521434106327455893405804, 6.63909590070426471146028025755, 7.45738154251477171724410622108, 7.938228196652915105973268773811
