| L(s) = 1 | + 0.741·2-s + 3-s − 1.45·4-s + 0.741·6-s + 1.03·7-s − 2.55·8-s + 9-s − 0.513·11-s − 1.45·12-s + 3.54·13-s + 0.767·14-s + 1.00·16-s − 1.36·17-s + 0.741·18-s + 0.894·19-s + 1.03·21-s − 0.380·22-s + 5.45·23-s − 2.55·24-s + 2.62·26-s + 27-s − 1.50·28-s − 9.65·29-s + 10.4·31-s + 5.86·32-s − 0.513·33-s − 1.01·34-s + ⋯ |
| L(s) = 1 | + 0.524·2-s + 0.577·3-s − 0.725·4-s + 0.302·6-s + 0.391·7-s − 0.904·8-s + 0.333·9-s − 0.154·11-s − 0.418·12-s + 0.982·13-s + 0.205·14-s + 0.251·16-s − 0.331·17-s + 0.174·18-s + 0.205·19-s + 0.226·21-s − 0.0812·22-s + 1.13·23-s − 0.522·24-s + 0.515·26-s + 0.192·27-s − 0.283·28-s − 1.79·29-s + 1.87·31-s + 1.03·32-s − 0.0894·33-s − 0.173·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.446760171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.446760171\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 0.741T + 2T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 - 0.894T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 7.99T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−9.160513906458760008774515023138, −8.485864402332870522344073261478, −7.85467279823171815719095604444, −6.82067882336286475599651542388, −5.85634333172795158745590765064, −5.06986982564824850134721437560, −4.20235780454735134368603365397, −3.50516289280285759703462661708, −2.48434612262146907690971297093, −1.00974510995344618287418762267,
1.00974510995344618287418762267, 2.48434612262146907690971297093, 3.50516289280285759703462661708, 4.20235780454735134368603365397, 5.06986982564824850134721437560, 5.85634333172795158745590765064, 6.82067882336286475599651542388, 7.85467279823171815719095604444, 8.485864402332870522344073261478, 9.160513906458760008774515023138
