| L(s) = 1 | + 0.874·2-s − 1.23·4-s − 7-s − 2.82·8-s + 0.333·11-s − 3.47·13-s − 0.874·14-s − 5.11·17-s − 5.23·19-s + 0.291·22-s + 8.81·23-s − 3.03·26-s + 1.23·28-s − 3.70·29-s − 3·31-s + 5.65·32-s − 4.47·34-s + 6.70·37-s − 4.57·38-s + 5.11·41-s + 5.76·43-s − 0.412·44-s + 7.70·46-s − 10.5·47-s + 49-s + 4.29·52-s + 11.2·53-s + ⋯ |
| L(s) = 1 | + 0.618·2-s − 0.618·4-s − 0.377·7-s − 0.999·8-s + 0.100·11-s − 0.962·13-s − 0.233·14-s − 1.24·17-s − 1.20·19-s + 0.0622·22-s + 1.83·23-s − 0.595·26-s + 0.233·28-s − 0.687·29-s − 0.538·31-s + 0.999·32-s − 0.766·34-s + 1.10·37-s − 0.742·38-s + 0.799·41-s + 0.878·43-s − 0.0622·44-s + 1.13·46-s − 1.54·47-s + 0.142·49-s + 0.595·52-s + 1.54·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.314888658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314888658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.874T + 2T^{2} \) |
| 11 | \( 1 - 0.333T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−8.428348623832356078527007088189, −7.45224995430918210756422899714, −6.73182572200669273701784127357, −6.06112007124857570407474695072, −5.19252728717438051103655802593, −4.56372839373375156049358739435, −3.95226716380957776637140885948, −2.96665099948059103351335290400, −2.20089372016998826278509058374, −0.55484243526708857145693600189,
0.55484243526708857145693600189, 2.20089372016998826278509058374, 2.96665099948059103351335290400, 3.95226716380957776637140885948, 4.56372839373375156049358739435, 5.19252728717438051103655802593, 6.06112007124857570407474695072, 6.73182572200669273701784127357, 7.45224995430918210756422899714, 8.428348623832356078527007088189
