| L(s) = 1 | + 5·2-s + 17·4-s − 5·5-s − 30·7-s + 45·8-s − 25·10-s + 50·11-s − 20·13-s − 150·14-s + 89·16-s − 10·17-s − 44·19-s − 85·20-s + 250·22-s + 120·23-s + 25·25-s − 100·26-s − 510·28-s − 50·29-s + 108·31-s + 85·32-s − 50·34-s + 150·35-s − 40·37-s − 220·38-s − 225·40-s + 400·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s − 0.447·5-s − 1.61·7-s + 1.98·8-s − 0.790·10-s + 1.37·11-s − 0.426·13-s − 2.86·14-s + 1.39·16-s − 0.142·17-s − 0.531·19-s − 0.950·20-s + 2.42·22-s + 1.08·23-s + 1/5·25-s − 0.754·26-s − 3.44·28-s − 0.320·29-s + 0.625·31-s + 0.469·32-s − 0.252·34-s + 0.724·35-s − 0.177·37-s − 0.939·38-s − 0.889·40-s + 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.831892324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831892324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 30 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 - 400 T + p^{3} T^{2} \) |
| 43 | \( 1 - 280 T + p^{3} T^{2} \) |
| 47 | \( 1 + 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 610 T + p^{3} T^{2} \) |
| 59 | \( 1 - 50 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 + 516 T + p^{3} T^{2} \) |
| 83 | \( 1 - 660 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1500 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1630 T + p^{3} T^{2} \) |
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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
−15.12556069456682899459309819912, −14.13379191546024720838265641441, −12.92377053573407567405289029338, −12.30612595013891763301803500977, −11.10229573471413830839720246939, −9.373493726583724709752712435148, −7.00197329551803109455818104635, −6.13920285501397724200463337114, −4.30204614990575151079004770500, −3.09368429162522383156924001001,
3.09368429162522383156924001001, 4.30204614990575151079004770500, 6.13920285501397724200463337114, 7.00197329551803109455818104635, 9.373493726583724709752712435148, 11.10229573471413830839720246939, 12.30612595013891763301803500977, 12.92377053573407567405289029338, 14.13379191546024720838265641441, 15.12556069456682899459309819912
