| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 11-s − 12-s − 3·13-s + 4·14-s + 16-s − 17-s − 18-s + 19-s + 4·21-s + 22-s − 23-s + 24-s + 3·26-s − 27-s − 4·28-s + 7·29-s − 9·31-s − 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.832·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s + 0.872·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.755·28-s + 1.29·29-s − 1.61·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−15.93888705302351, −15.45783613457626, −14.87103426180708, −14.15924841908303, −13.52746371888503, −12.83163597108564, −12.62444974228503, −11.92217896870497, −11.53144827551301, −10.71416658088537, −10.27520885162064, −9.801347542488444, −9.452428684415750, −8.695273676064187, −8.159647708576565, −7.298732679744316, −6.878761170524209, −6.515242821939969, −5.727646685779535, −5.221140237985990, −4.434775943163297, −3.495782409147703, −3.023473404588401, −2.205844296188321, −1.322016266453490, 0, 0,
1.322016266453490, 2.205844296188321, 3.023473404588401, 3.495782409147703, 4.434775943163297, 5.221140237985990, 5.727646685779535, 6.515242821939969, 6.878761170524209, 7.298732679744316, 8.159647708576565, 8.695273676064187, 9.452428684415750, 9.801347542488444, 10.27520885162064, 10.71416658088537, 11.53144827551301, 11.92217896870497, 12.62444974228503, 12.83163597108564, 13.52746371888503, 14.15924841908303, 14.87103426180708, 15.45783613457626, 15.93888705302351
