L(s) = 1 | + 2.02·2-s − 2.05·3-s + 2.10·4-s + 5-s − 4.17·6-s − 1.09·7-s + 0.215·8-s + 1.24·9-s + 2.02·10-s + 4.99·11-s − 4.33·12-s − 3.96·13-s − 2.22·14-s − 2.05·15-s − 3.77·16-s + 17-s + 2.51·18-s − 3.83·19-s + 2.10·20-s + 2.25·21-s + 10.1·22-s + 5.83·23-s − 0.444·24-s + 25-s − 8.04·26-s + 3.62·27-s − 2.30·28-s + ⋯ |
L(s) = 1 | + 1.43·2-s − 1.18·3-s + 1.05·4-s + 0.447·5-s − 1.70·6-s − 0.414·7-s + 0.0762·8-s + 0.413·9-s + 0.640·10-s + 1.50·11-s − 1.25·12-s − 1.10·13-s − 0.593·14-s − 0.531·15-s − 0.943·16-s + 0.242·17-s + 0.592·18-s − 0.880·19-s + 0.471·20-s + 0.492·21-s + 2.15·22-s + 1.21·23-s − 0.0906·24-s + 0.200·25-s − 1.57·26-s + 0.697·27-s − 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 - 4.99T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.27T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 7.68T + 61T^{2} \) |
| 67 | \( 1 + 5.08T + 67T^{2} \) |
| 73 | \( 1 + 3.02T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
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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
−6.99412520611070709826733669721, −6.82654275343536717752070476173, −6.04111734717360505913423939281, −5.63404106031470438414112845638, −4.76932189963567172517143137918, −4.40136336296633132156408858063, −3.38337796744925826262896845738, −2.61939696571985156350819002457, −1.41272799922776531509082896018, 0,
1.41272799922776531509082896018, 2.61939696571985156350819002457, 3.38337796744925826262896845738, 4.40136336296633132156408858063, 4.76932189963567172517143137918, 5.63404106031470438414112845638, 6.04111734717360505913423939281, 6.82654275343536717752070476173, 6.99412520611070709826733669721