| L(s) = 1 | − 1.30·2-s + 3-s − 0.302·4-s − 1.30·6-s + 7-s + 3·8-s + 9-s − 3·11-s − 0.302·12-s + 4.60·13-s − 1.30·14-s − 3.30·16-s − 2.60·17-s − 1.30·18-s − 0.605·19-s + 21-s + 3.90·22-s + 8.21·23-s + 3·24-s − 6·26-s + 27-s − 0.302·28-s − 0.394·29-s + 7.21·31-s − 1.69·32-s − 3·33-s + 3.39·34-s + ⋯ |
| L(s) = 1 | − 0.921·2-s + 0.577·3-s − 0.151·4-s − 0.531·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s − 0.904·11-s − 0.0874·12-s + 1.27·13-s − 0.348·14-s − 0.825·16-s − 0.631·17-s − 0.307·18-s − 0.138·19-s + 0.218·21-s + 0.833·22-s + 1.71·23-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 0.0572·28-s − 0.0732·29-s + 1.29·31-s − 0.300·32-s − 0.522·33-s + 0.582·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050980385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050980385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 + 0.394T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 6.60T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−10.79780919306747793069880154372, −9.793335438532053116002002357886, −8.966703555107316466281600066345, −8.318211282096011046886671201429, −7.67393818503279158702144974855, −6.54179657572404628384987199966, −5.10049332802200492366671889537, −4.12608966068775967928571524911, −2.65940509197039901278093412004, −1.12404555162790709900654244864,
1.12404555162790709900654244864, 2.65940509197039901278093412004, 4.12608966068775967928571524911, 5.10049332802200492366671889537, 6.54179657572404628384987199966, 7.67393818503279158702144974855, 8.318211282096011046886671201429, 8.966703555107316466281600066345, 9.793335438532053116002002357886, 10.79780919306747793069880154372
