| L(s) = 1 | − 2.52·2-s − 3.25·3-s + 4.38·4-s − 2.46·5-s + 8.21·6-s − 3.38·7-s − 6.01·8-s + 7.58·9-s + 6.23·10-s + 4.40·11-s − 14.2·12-s + 13-s + 8.55·14-s + 8.02·15-s + 6.42·16-s + 1.63·17-s − 19.1·18-s + 4.16·19-s − 10.8·20-s + 11.0·21-s − 11.1·22-s + 1.46·23-s + 19.5·24-s + 1.08·25-s − 2.52·26-s − 14.9·27-s − 14.8·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.87·3-s + 2.19·4-s − 1.10·5-s + 3.35·6-s − 1.28·7-s − 2.12·8-s + 2.52·9-s + 1.97·10-s + 1.32·11-s − 4.11·12-s + 0.277·13-s + 2.28·14-s + 2.07·15-s + 1.60·16-s + 0.396·17-s − 4.51·18-s + 0.956·19-s − 2.41·20-s + 2.40·21-s − 2.37·22-s + 0.305·23-s + 3.99·24-s + 0.217·25-s − 0.495·26-s − 2.86·27-s − 2.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 0.120T + 31T^{2} \) |
| 37 | \( 1 - 9.09T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 - 1.67T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 8.90T + 97T^{2} \) |
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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
−7.36132616472882392481839351143, −6.95725028783865049550573852876, −6.20624965330855234381323085155, −5.99908006455450174841748474428, −4.70821658142493027563791845281, −3.88308182247959857889583627747, −3.04274719851594088334167100312, −1.41809798646518917453187516575, −0.813845544399873782427240950478, 0,
0.813845544399873782427240950478, 1.41809798646518917453187516575, 3.04274719851594088334167100312, 3.88308182247959857889583627747, 4.70821658142493027563791845281, 5.99908006455450174841748474428, 6.20624965330855234381323085155, 6.95725028783865049550573852876, 7.36132616472882392481839351143
