| L(s) = 1 | + 2.58·2-s − 3-s + 4.66·4-s + 5-s − 2.58·6-s + 0.946·7-s + 6.88·8-s + 9-s + 2.58·10-s − 0.528·11-s − 4.66·12-s − 7.01·13-s + 2.44·14-s − 15-s + 8.43·16-s + 2.30·17-s + 2.58·18-s + 4·19-s + 4.66·20-s − 0.946·21-s − 1.36·22-s − 4.46·23-s − 6.88·24-s + 25-s − 18.1·26-s − 27-s + 4.41·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.577·3-s + 2.33·4-s + 0.447·5-s − 1.05·6-s + 0.357·7-s + 2.43·8-s + 0.333·9-s + 0.816·10-s − 0.159·11-s − 1.34·12-s − 1.94·13-s + 0.653·14-s − 0.258·15-s + 2.10·16-s + 0.557·17-s + 0.608·18-s + 0.917·19-s + 1.04·20-s − 0.206·21-s − 0.291·22-s − 0.931·23-s − 1.40·24-s + 0.200·25-s − 3.55·26-s − 0.192·27-s + 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.541986194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.541986194\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 7 | \( 1 - 0.946T + 7T^{2} \) |
| 11 | \( 1 + 0.528T + 11T^{2} \) |
| 13 | \( 1 + 7.01T + 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 0.863T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 + 1.47T + 97T^{2} \) |
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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
−11.47694493354230151279548166658, −10.41303821044900942045648942129, −9.666493071177756674467961815472, −7.77999332001281975653138138238, −7.02657973620908077679462760817, −5.98703412879406635404465388987, −5.16369197364004096883377848812, −4.61274457673710009836924025056, −3.18990048675742561578658892536, −1.99734687363675633370292512642,
1.99734687363675633370292512642, 3.18990048675742561578658892536, 4.61274457673710009836924025056, 5.16369197364004096883377848812, 5.98703412879406635404465388987, 7.02657973620908077679462760817, 7.77999332001281975653138138238, 9.666493071177756674467961815472, 10.41303821044900942045648942129, 11.47694493354230151279548166658
