| L(s) = 1 | + 0.732·2-s − 3-s − 1.46·4-s + 5-s − 0.732·6-s − 2.53·8-s + 9-s + 0.732·10-s − 2.73·11-s + 1.46·12-s + 5.73·13-s − 15-s + 1.07·16-s + 6.73·17-s + 0.732·18-s − 2.46·19-s − 1.46·20-s − 2·22-s − 1.26·23-s + 2.53·24-s + 25-s + 4.19·26-s − 27-s + 6.19·29-s − 0.732·30-s + 6.46·31-s + 5.85·32-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.577·3-s − 0.732·4-s + 0.447·5-s − 0.298·6-s − 0.896·8-s + 0.333·9-s + 0.231·10-s − 0.823·11-s + 0.422·12-s + 1.58·13-s − 0.258·15-s + 0.267·16-s + 1.63·17-s + 0.172·18-s − 0.565·19-s − 0.327·20-s − 0.426·22-s − 0.264·23-s + 0.517·24-s + 0.200·25-s + 0.822·26-s − 0.192·27-s + 1.15·29-s − 0.133·30-s + 1.16·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464129340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464129340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−10.27036769809479083075884081211, −9.778186748190145099850094740111, −8.554808346617920577059426734340, −7.981246231671966534200358295995, −6.44317236645923943628425439163, −5.83271814970507657037252491399, −5.05242130557099098677091489291, −4.04531268071808069611300017411, −2.95071109530728965709677561584, −1.02739748126678866703037333369,
1.02739748126678866703037333369, 2.95071109530728965709677561584, 4.04531268071808069611300017411, 5.05242130557099098677091489291, 5.83271814970507657037252491399, 6.44317236645923943628425439163, 7.981246231671966534200358295995, 8.554808346617920577059426734340, 9.778186748190145099850094740111, 10.27036769809479083075884081211
