| L(s) = 1 | + 1.26·2-s − 3-s − 0.392·4-s − 5-s − 1.26·6-s − 4.11·7-s − 3.03·8-s + 9-s − 1.26·10-s − 0.250·11-s + 0.392·12-s + 1.98·13-s − 5.21·14-s + 15-s − 3.06·16-s − 4.97·17-s + 1.26·18-s − 5.64·19-s + 0.392·20-s + 4.11·21-s − 0.318·22-s − 7.05·23-s + 3.03·24-s + 25-s + 2.52·26-s − 27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.896·2-s − 0.577·3-s − 0.196·4-s − 0.447·5-s − 0.517·6-s − 1.55·7-s − 1.07·8-s + 0.333·9-s − 0.400·10-s − 0.0756·11-s + 0.113·12-s + 0.551·13-s − 1.39·14-s + 0.258·15-s − 0.765·16-s − 1.20·17-s + 0.298·18-s − 1.29·19-s + 0.0876·20-s + 0.897·21-s − 0.0678·22-s − 1.47·23-s + 0.619·24-s + 0.200·25-s + 0.494·26-s − 0.192·27-s + 0.304·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2382019449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2382019449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 0.250T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 0.118T + 37T^{2} \) |
| 41 | \( 1 + 0.127T + 41T^{2} \) |
| 43 | \( 1 + 1.69T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + 6.11T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 4.62T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 4.20T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−8.080314437480351877613277107114, −7.01233267871486656766117013600, −6.34961073582587017609661861809, −6.07494734745024878488674137989, −5.20409683808023706064944083710, −4.22383494330653178561274153443, −3.91262146377317749489177347222, −3.11732073343371748507612727661, −2.06398403858155046285495705366, −0.21480744179225732275260708451,
0.21480744179225732275260708451, 2.06398403858155046285495705366, 3.11732073343371748507612727661, 3.91262146377317749489177347222, 4.22383494330653178561274153443, 5.20409683808023706064944083710, 6.07494734745024878488674137989, 6.34961073582587017609661861809, 7.01233267871486656766117013600, 8.080314437480351877613277107114
