| L(s) = 1 | + 3.23·5-s + 7-s + 3.23·11-s − 4.47·13-s − 0.763·17-s − 2.47·19-s + 5.70·23-s + 5.47·25-s + 1.52·29-s − 2.47·31-s + 3.23·35-s + 4.47·37-s − 7.23·41-s + 12.9·43-s + 1.52·47-s + 49-s + 8·53-s + 10.4·55-s + 6.47·59-s + 4.47·61-s − 14.4·65-s + 8·67-s + 7.23·71-s − 14.9·73-s + 3.23·77-s + 12.9·79-s − 12.9·83-s + ⋯ |
| L(s) = 1 | + 1.44·5-s + 0.377·7-s + 0.975·11-s − 1.24·13-s − 0.185·17-s − 0.567·19-s + 1.19·23-s + 1.09·25-s + 0.283·29-s − 0.444·31-s + 0.546·35-s + 0.735·37-s − 1.13·41-s + 1.97·43-s + 0.222·47-s + 0.142·49-s + 1.09·53-s + 1.41·55-s + 0.842·59-s + 0.572·61-s − 1.79·65-s + 0.977·67-s + 0.858·71-s − 1.74·73-s + 0.368·77-s + 1.45·79-s − 1.42·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.800959160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800959160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.768137402244535847866796932088, −7.57060651306954935280200700049, −6.89979256518111088045281614264, −6.24878460444227487272859827261, −5.44847645574987078954552426945, −4.83313134233721954403699952949, −3.92996662271102959429877738833, −2.64653201428946196027156339575, −2.06166184904272634594278488980, −1.00424009298909905885542056993,
1.00424009298909905885542056993, 2.06166184904272634594278488980, 2.64653201428946196027156339575, 3.92996662271102959429877738833, 4.83313134233721954403699952949, 5.44847645574987078954552426945, 6.24878460444227487272859827261, 6.89979256518111088045281614264, 7.57060651306954935280200700049, 8.768137402244535847866796932088
