| L(s) = 1 | + 3·2-s + 4-s + 5·5-s + 7·7-s − 21·8-s + 15·10-s + 60·11-s + 38·13-s + 21·14-s − 71·16-s − 84·17-s + 110·19-s + 5·20-s + 180·22-s + 120·23-s + 25·25-s + 114·26-s + 7·28-s + 162·29-s + 236·31-s − 45·32-s − 252·34-s + 35·35-s − 376·37-s + 330·38-s − 105·40-s − 126·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 0.447·5-s + 0.377·7-s − 0.928·8-s + 0.474·10-s + 1.64·11-s + 0.810·13-s + 0.400·14-s − 1.10·16-s − 1.19·17-s + 1.32·19-s + 0.0559·20-s + 1.74·22-s + 1.08·23-s + 1/5·25-s + 0.859·26-s + 0.0472·28-s + 1.03·29-s + 1.36·31-s − 0.248·32-s − 1.27·34-s + 0.169·35-s − 1.67·37-s + 1.40·38-s − 0.415·40-s − 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.520292536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520292536\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 376 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 + 6 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 880 T + p^{3} T^{2} \) |
| 67 | \( 1 + 826 T + p^{3} T^{2} \) |
| 71 | \( 1 + 666 T + p^{3} T^{2} \) |
| 73 | \( 1 + 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 - 792 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1002 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1442 T + p^{3} T^{2} \) |
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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
−11.70539205514704003597646040648, −10.43193107180795739663972008163, −9.106395941494474397243163013178, −8.720191915741096461651970571706, −6.91786986868751059335531761891, −6.20093816968590882556250996210, −5.06532400364823597101877458716, −4.15366849174035422445509180431, −3.02796480050487528963941092002, −1.26757353380453565560981955821,
1.26757353380453565560981955821, 3.02796480050487528963941092002, 4.15366849174035422445509180431, 5.06532400364823597101877458716, 6.20093816968590882556250996210, 6.91786986868751059335531761891, 8.720191915741096461651970571706, 9.106395941494474397243163013178, 10.43193107180795739663972008163, 11.70539205514704003597646040648
