| L(s) = 1 | + 1.61·2-s − 1.83·3-s + 0.623·4-s − 5-s − 2.97·6-s − 7-s − 2.22·8-s + 0.362·9-s − 1.61·10-s + 4.22·11-s − 1.14·12-s + 4.23·13-s − 1.61·14-s + 1.83·15-s − 4.85·16-s − 6.13·17-s + 0.587·18-s − 0.394·19-s − 0.623·20-s + 1.83·21-s + 6.83·22-s − 1.77·23-s + 4.08·24-s + 25-s + 6.86·26-s + 4.83·27-s − 0.623·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.05·3-s + 0.311·4-s − 0.447·5-s − 1.21·6-s − 0.377·7-s − 0.788·8-s + 0.120·9-s − 0.512·10-s + 1.27·11-s − 0.330·12-s + 1.17·13-s − 0.432·14-s + 0.473·15-s − 1.21·16-s − 1.48·17-s + 0.138·18-s − 0.0904·19-s − 0.139·20-s + 0.400·21-s + 1.45·22-s − 0.369·23-s + 0.834·24-s + 0.200·25-s + 1.34·26-s + 0.930·27-s − 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.83T + 3T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 + 0.394T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 - 3.87T + 43T^{2} \) |
| 47 | \( 1 + 2.25T + 47T^{2} \) |
| 53 | \( 1 + 1.25T + 53T^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 - 9.60T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 0.644T + 71T^{2} \) |
| 73 | \( 1 + 5.71T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
| 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
−6.88290532019013167896392221909, −6.56364099517321981313268899606, −6.10998660195651467843066250394, −5.38592258573536111205336334346, −4.58829131339949540026825649187, −4.06947057088486303228459458165, −3.47434018771921566559551287092, −2.49228653977446755030596599473, −1.10319887591876089086611550824, 0,
1.10319887591876089086611550824, 2.49228653977446755030596599473, 3.47434018771921566559551287092, 4.06947057088486303228459458165, 4.58829131339949540026825649187, 5.38592258573536111205336334346, 6.10998660195651467843066250394, 6.56364099517321981313268899606, 6.88290532019013167896392221909
