| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 3·11-s + 12-s + 13-s − 14-s + 16-s − 18-s + 5·19-s + 21-s + 3·22-s + 3·23-s − 24-s − 5·25-s − 26-s + 27-s + 28-s + 5·31-s − 32-s − 3·33-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.14·19-s + 0.218·21-s + 0.639·22-s + 0.625·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.898·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077793774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077793774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.66381546543225, −14.85403923844523, −14.74695978822788, −13.70214024073331, −13.50257397846806, −12.92840987015007, −12.06087335463028, −11.66418944263077, −11.08593461709795, −10.42794467947425, −9.888496953870859, −9.519639610709437, −8.725406539304806, −8.230062429317305, −7.851307159002200, −7.186649896502729, −6.695115263069576, −5.716710895435779, −5.288133359812062, −4.455944546180457, −3.636572303235739, −2.926070488817589, −2.327910195069931, −1.487053930330593, −0.6520447829128608,
0.6520447829128608, 1.487053930330593, 2.327910195069931, 2.926070488817589, 3.636572303235739, 4.455944546180457, 5.288133359812062, 5.716710895435779, 6.695115263069576, 7.186649896502729, 7.851307159002200, 8.230062429317305, 8.725406539304806, 9.519639610709437, 9.888496953870859, 10.42794467947425, 11.08593461709795, 11.66418944263077, 12.06087335463028, 12.92840987015007, 13.50257397846806, 13.70214024073331, 14.74695978822788, 14.85403923844523, 15.66381546543225
