| L(s) = 1 | + 1.94·2-s + 2.74·3-s + 1.76·4-s − 0.683·5-s + 5.32·6-s − 0.456·8-s + 4.52·9-s − 1.32·10-s + 0.192·11-s + 4.83·12-s + 6.05·13-s − 1.87·15-s − 4.41·16-s − 1.81·17-s + 8.77·18-s + 19-s − 1.20·20-s + 0.373·22-s + 2.34·23-s − 1.25·24-s − 4.53·25-s + 11.7·26-s + 4.17·27-s − 4.90·29-s − 3.63·30-s + 4.62·31-s − 7.65·32-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 1.58·3-s + 0.882·4-s − 0.305·5-s + 2.17·6-s − 0.161·8-s + 1.50·9-s − 0.419·10-s + 0.0580·11-s + 1.39·12-s + 1.67·13-s − 0.483·15-s − 1.10·16-s − 0.439·17-s + 2.06·18-s + 0.229·19-s − 0.269·20-s + 0.0795·22-s + 0.487·23-s − 0.255·24-s − 0.906·25-s + 2.30·26-s + 0.802·27-s − 0.910·29-s − 0.664·30-s + 0.830·31-s − 1.35·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.930536059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.930536059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 + 0.683T + 5T^{2} \) |
| 11 | \( 1 - 0.192T + 11T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 8.35T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 - 4.43T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + 8.38T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.971799897355650938172332487739, −8.907313800821028682934263661796, −8.555987068956880583473467785606, −7.51380352192144952120662203020, −6.56070253766898962171587268970, −5.59034743623176139552298543931, −4.36123191404854325742700046879, −3.66279171638060080550719821702, −3.07765492475641199939382969365, −1.84345601469911651156686421691,
1.84345601469911651156686421691, 3.07765492475641199939382969365, 3.66279171638060080550719821702, 4.36123191404854325742700046879, 5.59034743623176139552298543931, 6.56070253766898962171587268970, 7.51380352192144952120662203020, 8.555987068956880583473467785606, 8.907313800821028682934263661796, 9.971799897355650938172332487739
