| L(s) = 1 | − 0.633·2-s − 1.59·4-s + 5-s − 1.79·7-s + 2.27·8-s − 0.633·10-s − 4.05·11-s + 13-s + 1.13·14-s + 1.75·16-s + 7.09·17-s − 1.65·19-s − 1.59·20-s + 2.56·22-s − 8.14·23-s + 25-s − 0.633·26-s + 2.86·28-s + 1.63·29-s + 4.34·31-s − 5.66·32-s − 4.48·34-s − 1.79·35-s + 0.729·37-s + 1.04·38-s + 2.27·40-s + 0.879·41-s + ⋯ |
| L(s) = 1 | − 0.447·2-s − 0.799·4-s + 0.447·5-s − 0.677·7-s + 0.805·8-s − 0.200·10-s − 1.22·11-s + 0.277·13-s + 0.303·14-s + 0.438·16-s + 1.71·17-s − 0.379·19-s − 0.357·20-s + 0.547·22-s − 1.69·23-s + 0.200·25-s − 0.124·26-s + 0.541·28-s + 0.303·29-s + 0.780·31-s − 1.00·32-s − 0.769·34-s − 0.303·35-s + 0.119·37-s + 0.170·38-s + 0.360·40-s + 0.137·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.633T + 2T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 + 4.05T + 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 8.14T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 0.729T + 37T^{2} \) |
| 41 | \( 1 - 0.879T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 - 5.39T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 - 4.37T + 67T^{2} \) |
| 71 | \( 1 + 2.80T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.171847690572039114354527549520, −7.32975166961116513610197473834, −6.28613184799209003990988433590, −5.65976010229770932119002105586, −5.03270531772354911758399111909, −4.07733605294998364101217471390, −3.29485602827002556847764625389, −2.33871817936652337147818055140, −1.13361795507537916435459604013, 0,
1.13361795507537916435459604013, 2.33871817936652337147818055140, 3.29485602827002556847764625389, 4.07733605294998364101217471390, 5.03270531772354911758399111909, 5.65976010229770932119002105586, 6.28613184799209003990988433590, 7.32975166961116513610197473834, 8.171847690572039114354527549520
