| L(s) = 1 | − 0.338·3-s + 0.413·5-s − 1.61·7-s − 2.88·9-s + 1.62·11-s + 1.42·13-s − 0.140·15-s + 0.598·17-s + 19-s + 0.545·21-s − 0.815·23-s − 4.82·25-s + 1.99·27-s + 7.29·29-s − 4.96·31-s − 0.549·33-s − 0.667·35-s + 5.22·37-s − 0.481·39-s + 2.10·41-s − 5.53·43-s − 1.19·45-s + 5.12·47-s − 4.40·49-s − 0.202·51-s − 5.32·53-s + 0.672·55-s + ⋯ |
| L(s) = 1 | − 0.195·3-s + 0.185·5-s − 0.609·7-s − 0.961·9-s + 0.489·11-s + 0.394·13-s − 0.0361·15-s + 0.145·17-s + 0.229·19-s + 0.118·21-s − 0.170·23-s − 0.965·25-s + 0.383·27-s + 1.35·29-s − 0.891·31-s − 0.0956·33-s − 0.112·35-s + 0.859·37-s − 0.0770·39-s + 0.329·41-s − 0.843·43-s − 0.178·45-s + 0.747·47-s − 0.629·49-s − 0.0283·51-s − 0.731·53-s + 0.0906·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.338T + 3T^{2} \) |
| 5 | \( 1 - 0.413T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 - 0.598T + 17T^{2} \) |
| 23 | \( 1 + 0.815T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 0.614T + 89T^{2} \) |
| 97 | \( 1 + 2.40T + 97T^{2} \) |
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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
−7.79916891783509350130993731148, −6.87461945935253747666519334598, −6.19885022889159293957892850301, −5.77506879673873772344217729896, −4.90956122100070754334533808853, −3.95636639231423588036277958947, −3.22482419154965977360306538323, −2.42683459748559173410240262332, −1.24582298967227430404226807957, 0,
1.24582298967227430404226807957, 2.42683459748559173410240262332, 3.22482419154965977360306538323, 3.95636639231423588036277958947, 4.90956122100070754334533808853, 5.77506879673873772344217729896, 6.19885022889159293957892850301, 6.87461945935253747666519334598, 7.79916891783509350130993731148
