| L(s) = 1 | + 1.38·2-s − 3-s − 0.0861·4-s − 5-s − 1.38·6-s − 0.337·7-s − 2.88·8-s + 9-s − 1.38·10-s + 4.92·11-s + 0.0861·12-s + 3.27·13-s − 0.466·14-s + 15-s − 3.82·16-s + 0.592·17-s + 1.38·18-s − 1.10·19-s + 0.0861·20-s + 0.337·21-s + 6.80·22-s + 1.88·23-s + 2.88·24-s + 25-s + 4.52·26-s − 27-s + 0.0290·28-s + ⋯ |
| L(s) = 1 | + 0.978·2-s − 0.577·3-s − 0.0430·4-s − 0.447·5-s − 0.564·6-s − 0.127·7-s − 1.02·8-s + 0.333·9-s − 0.437·10-s + 1.48·11-s + 0.0248·12-s + 0.907·13-s − 0.124·14-s + 0.258·15-s − 0.955·16-s + 0.143·17-s + 0.326·18-s − 0.254·19-s + 0.0192·20-s + 0.0735·21-s + 1.45·22-s + 0.392·23-s + 0.589·24-s + 0.200·25-s + 0.887·26-s − 0.192·27-s + 0.00549·28-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
| \(L(1)\) |
\(\approx\) |
\(2.085641834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085641834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.38T + 2T^{2} \) |
| 7 | \( 1 + 0.337T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 - 0.592T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 5.45T + 73T^{2} \) |
| 79 | \( 1 - 5.60T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 + 0.957T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.146904772297661607014774287725, −6.93542968797718101816119197281, −6.57612585524158262324711936451, −5.86624863543572843335574729486, −5.14886613046923803136588195125, −4.41564523621501308027303592662, −3.69319733066369076805891378235, −3.34017746858252405108880215220, −1.83721671578625479522155081025, −0.68914806139634220069092735371,
0.68914806139634220069092735371, 1.83721671578625479522155081025, 3.34017746858252405108880215220, 3.69319733066369076805891378235, 4.41564523621501308027303592662, 5.14886613046923803136588195125, 5.86624863543572843335574729486, 6.57612585524158262324711936451, 6.93542968797718101816119197281, 8.146904772297661607014774287725
