| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 5·11-s − 12-s − 15-s + 16-s − 4·17-s + 18-s + 8·19-s + 20-s + 5·22-s − 4·23-s − 24-s − 4·25-s − 27-s − 5·29-s − 30-s + 3·31-s + 32-s − 5·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.06·22-s − 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.928·29-s − 0.182·30-s + 0.538·31-s + 0.176·32-s − 0.870·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828000428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828000428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−11.74523480292824260887423342382, −11.22224988296755911606023262749, −9.923651152419313062412512140731, −9.180591929632892398823763196977, −7.62779471882474450110803429137, −6.55776248876886813154572094683, −5.83426012328410247998142577853, −4.67074113127017439545556449138, −3.52909847925109475796916640798, −1.68608286850308687773629845560,
1.68608286850308687773629845560, 3.52909847925109475796916640798, 4.67074113127017439545556449138, 5.83426012328410247998142577853, 6.55776248876886813154572094683, 7.62779471882474450110803429137, 9.180591929632892398823763196977, 9.923651152419313062412512140731, 11.22224988296755911606023262749, 11.74523480292824260887423342382
