| L(s) = 1 | − 2.19·2-s + 0.358·3-s + 2.80·4-s + 5-s − 0.785·6-s − 1.77·8-s − 2.87·9-s − 2.19·10-s − 2.01·11-s + 1.00·12-s + 1.14·13-s + 0.358·15-s − 1.72·16-s − 3.27·17-s + 6.29·18-s + 4.79·19-s + 2.80·20-s + 4.42·22-s + 4.15·23-s − 0.635·24-s + 25-s − 2.50·26-s − 2.10·27-s + 29-s − 0.785·30-s + 2.36·31-s + 7.33·32-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.206·3-s + 1.40·4-s + 0.447·5-s − 0.320·6-s − 0.627·8-s − 0.957·9-s − 0.693·10-s − 0.607·11-s + 0.290·12-s + 0.317·13-s + 0.0924·15-s − 0.431·16-s − 0.794·17-s + 1.48·18-s + 1.09·19-s + 0.628·20-s + 0.942·22-s + 0.865·23-s − 0.129·24-s + 0.200·25-s − 0.491·26-s − 0.404·27-s + 0.185·29-s − 0.143·30-s + 0.425·31-s + 1.29·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8074908039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8074908039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 0.358T + 3T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 + 0.0951T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 8.10T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 + 2.05T + 89T^{2} \) |
| 97 | \( 1 + 5.56T + 97T^{2} \) |
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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
−8.104694437005733265801489703640, −7.43921310217601145415803197532, −6.83299149965312188943282777774, −5.97883139620296565574145791357, −5.31789665960587835021349885211, −4.38995895931247556628212064892, −3.07798364272026420577541164909, −2.56080524961508555510550819616, −1.58684568683845597145540022710, −0.57981246563541844727953707936,
0.57981246563541844727953707936, 1.58684568683845597145540022710, 2.56080524961508555510550819616, 3.07798364272026420577541164909, 4.38995895931247556628212064892, 5.31789665960587835021349885211, 5.97883139620296565574145791357, 6.83299149965312188943282777774, 7.43921310217601145415803197532, 8.104694437005733265801489703640
