L(s) = 1 | + 2.32·2-s + 3-s + 3.38·4-s + 5-s + 2.32·6-s − 2.23·7-s + 3.21·8-s + 9-s + 2.32·10-s − 3.62·11-s + 3.38·12-s − 5.10·13-s − 5.18·14-s + 15-s + 0.695·16-s − 4.51·17-s + 2.32·18-s − 6.48·19-s + 3.38·20-s − 2.23·21-s − 8.41·22-s − 3.83·23-s + 3.21·24-s + 25-s − 11.8·26-s + 27-s − 7.56·28-s + ⋯ |
L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.69·4-s + 0.447·5-s + 0.947·6-s − 0.844·7-s + 1.13·8-s + 0.333·9-s + 0.733·10-s − 1.09·11-s + 0.977·12-s − 1.41·13-s − 1.38·14-s + 0.258·15-s + 0.173·16-s − 1.09·17-s + 0.547·18-s − 1.48·19-s + 0.757·20-s − 0.487·21-s − 1.79·22-s − 0.799·23-s + 0.656·24-s + 0.200·25-s − 2.32·26-s + 0.192·27-s − 1.43·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 2.32T + 2T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 - 8.52T + 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 - 0.448T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 1.62T + 73T^{2} \) |
| 79 | \( 1 + 6.87T + 79T^{2} \) |
| 83 | \( 1 - 1.24T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 5.50T + 97T^{2} \) |
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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
−7.51030128536603504318386209374, −6.62337449917663005628819617627, −6.34005759466163227679189814017, −5.46569694572007266692395021069, −4.53782040972199351392525162300, −4.35862723460119437479058245053, −3.12455080194949364799003818583, −2.54081354554030346352462097980, −2.14807778313415925930050076441, 0,
2.14807778313415925930050076441, 2.54081354554030346352462097980, 3.12455080194949364799003818583, 4.35862723460119437479058245053, 4.53782040972199351392525162300, 5.46569694572007266692395021069, 6.34005759466163227679189814017, 6.62337449917663005628819617627, 7.51030128536603504318386209374