| L(s) = 1 | + 1.42·2-s + 0.0167·4-s − 5-s + 3.66·7-s − 2.81·8-s − 1.42·10-s − 3.26·11-s − 2.79·13-s + 5.20·14-s − 4.03·16-s − 2.12·17-s + 6.67·19-s − 0.0167·20-s − 4.63·22-s + 1.64·23-s + 25-s − 3.97·26-s + 0.0614·28-s − 0.0189·29-s − 10.5·31-s − 0.0948·32-s − 3.01·34-s − 3.66·35-s − 1.71·37-s + 9.47·38-s + 2.81·40-s − 1.81·41-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.00838·4-s − 0.447·5-s + 1.38·7-s − 0.995·8-s − 0.449·10-s − 0.984·11-s − 0.776·13-s + 1.38·14-s − 1.00·16-s − 0.514·17-s + 1.53·19-s − 0.00375·20-s − 0.988·22-s + 0.342·23-s + 0.200·25-s − 0.779·26-s + 0.0116·28-s − 0.00351·29-s − 1.90·31-s − 0.0167·32-s − 0.516·34-s − 0.619·35-s − 0.281·37-s + 1.53·38-s + 0.445·40-s − 0.283·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 - 1.42T + 2T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 0.0189T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 71 | \( 1 - 8.11T + 71T^{2} \) |
| 73 | \( 1 - 5.96T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−8.163709971439548474347682346323, −7.63912942384398153122011370107, −6.86888484332353877186175639322, −5.64040272514203558098879723456, −4.98289131750586507240215832427, −4.78056646374565368105800219641, −3.63768212220956092532262337976, −2.86821213272054142341002202239, −1.71452398947341989805700178649, 0,
1.71452398947341989805700178649, 2.86821213272054142341002202239, 3.63768212220956092532262337976, 4.78056646374565368105800219641, 4.98289131750586507240215832427, 5.64040272514203558098879723456, 6.86888484332353877186175639322, 7.63912942384398153122011370107, 8.163709971439548474347682346323
