| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 7-s − 8-s + 6·9-s − 4·11-s − 3·12-s + 3·13-s + 14-s + 16-s + 17-s − 6·18-s + 6·19-s + 3·21-s + 4·22-s + 3·24-s − 3·26-s − 9·27-s − 28-s − 9·31-s − 32-s + 12·33-s − 34-s + 6·36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 1.20·11-s − 0.866·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 1.37·19-s + 0.654·21-s + 0.852·22-s + 0.612·24-s − 0.588·26-s − 1.73·27-s − 0.188·28-s − 1.61·31-s − 0.176·32-s + 2.08·33-s − 0.171·34-s + 36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.912079858561699963383102104798, −9.194075524487659594092066631529, −7.79245125853043497524015802224, −7.25725380166347220079989594968, −6.06625119903536027210672312295, −5.69252691577739764158095209578, −4.63606983114592049159737766087, −3.14363225178080646361043609520, −1.34569919885542900950402397627, 0,
1.34569919885542900950402397627, 3.14363225178080646361043609520, 4.63606983114592049159737766087, 5.69252691577739764158095209578, 6.06625119903536027210672312295, 7.25725380166347220079989594968, 7.79245125853043497524015802224, 9.194075524487659594092066631529, 9.912079858561699963383102104798
