| L(s) = 1 | + 2.40·2-s + 1.94·3-s + 3.78·4-s + 4.68·6-s − 0.665·7-s + 4.28·8-s + 0.794·9-s + 11-s + 7.36·12-s + 3.05·13-s − 1.59·14-s + 2.73·16-s + 4.49·17-s + 1.91·18-s + 19-s − 1.29·21-s + 2.40·22-s + 0.205·23-s + 8.34·24-s + 7.34·26-s − 4.29·27-s − 2.51·28-s + 0.0417·29-s + 6.76·31-s − 1.98·32-s + 1.94·33-s + 10.8·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.12·3-s + 1.89·4-s + 1.91·6-s − 0.251·7-s + 1.51·8-s + 0.264·9-s + 0.301·11-s + 2.12·12-s + 0.847·13-s − 0.427·14-s + 0.683·16-s + 1.08·17-s + 0.450·18-s + 0.229·19-s − 0.282·21-s + 0.512·22-s + 0.0427·23-s + 1.70·24-s + 1.44·26-s − 0.826·27-s − 0.475·28-s + 0.00776·29-s + 1.21·31-s − 0.351·32-s + 0.339·33-s + 1.85·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.711603013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.711603013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 - 1.94T + 3T^{2} \) |
| 7 | \( 1 + 0.665T + 7T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 23 | \( 1 - 0.205T + 23T^{2} \) |
| 29 | \( 1 - 0.0417T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 7.55T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 6.06T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
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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
−7.960490005246633530200480165702, −7.51758806371555721984659380455, −6.37872626274492668439207894176, −6.09401848432568600196876795821, −5.16987283543241482642315138717, −4.34288478887783214877913549269, −3.61080796841016430494597906193, −3.11313160669686981201942092390, −2.44870774196983349879227527548, −1.31796405591483271314376179127,
1.31796405591483271314376179127, 2.44870774196983349879227527548, 3.11313160669686981201942092390, 3.61080796841016430494597906193, 4.34288478887783214877913549269, 5.16987283543241482642315138717, 6.09401848432568600196876795821, 6.37872626274492668439207894176, 7.51758806371555721984659380455, 7.960490005246633530200480165702
