| L(s) = 1 | − 1.23·2-s − 3-s − 0.477·4-s + 5-s + 1.23·6-s − 3.79·7-s + 3.05·8-s + 9-s − 1.23·10-s + 0.477·12-s − 2.46·13-s + 4.68·14-s − 15-s − 2.81·16-s + 6.52·17-s − 1.23·18-s − 8.25·19-s − 0.477·20-s + 3.79·21-s − 2.34·23-s − 3.05·24-s + 25-s + 3.04·26-s − 27-s + 1.81·28-s − 2.46·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | − 0.872·2-s − 0.577·3-s − 0.238·4-s + 0.447·5-s + 0.503·6-s − 1.43·7-s + 1.08·8-s + 0.333·9-s − 0.390·10-s + 0.137·12-s − 0.684·13-s + 1.25·14-s − 0.258·15-s − 0.704·16-s + 1.58·17-s − 0.290·18-s − 1.89·19-s − 0.106·20-s + 0.827·21-s − 0.487·23-s − 0.623·24-s + 0.200·25-s + 0.597·26-s − 0.192·27-s + 0.342·28-s − 0.458·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4596960349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4596960349\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 8.25T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.66T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.560100425017046171604412744327, −8.543975652148080775790915123991, −7.83813450969496069213487625356, −6.81270755357276208768078204530, −6.25546732118144264001826545816, −5.31643890216811223892405705180, −4.36371412397755383991477360608, −3.30495458457395109417196497002, −1.96760625177718074842108364427, −0.53106185488785757090461552161,
0.53106185488785757090461552161, 1.96760625177718074842108364427, 3.30495458457395109417196497002, 4.36371412397755383991477360608, 5.31643890216811223892405705180, 6.25546732118144264001826545816, 6.81270755357276208768078204530, 7.83813450969496069213487625356, 8.543975652148080775790915123991, 9.560100425017046171604412744327
