| L(s) = 1 | − 3-s − 2·4-s − 5-s + 2·7-s + 9-s + 4·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s − 7·17-s + 5·19-s + 2·20-s − 2·21-s + 23-s + 25-s − 27-s − 4·28-s − 8·31-s − 4·33-s − 2·35-s − 2·36-s + 37-s + 5·39-s − 4·41-s + 43-s − 8·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 1.69·17-s + 1.14·19-s + 0.447·20-s − 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.43·31-s − 0.696·33-s − 0.338·35-s − 1/3·36-s + 0.164·37-s + 0.800·39-s − 0.624·41-s + 0.152·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4350664125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4350664125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.69078652742848, −12.20084659583979, −11.74686563893708, −11.45845960004852, −10.96738689688702, −10.52568601492658, −9.800477148852323, −9.423414161808490, −9.156620307468419, −8.645918749100543, −7.977970595691556, −7.679826469475099, −7.080259040490634, −6.662082855633976, −6.142190051728950, −5.342500803879590, −4.995422159259445, −4.680320341053220, −4.221297961772802, −3.617559217662347, −3.161613946654456, −2.193636950791387, −1.659594324016829, −1.023784770386337, −0.2126945748968181,
0.2126945748968181, 1.023784770386337, 1.659594324016829, 2.193636950791387, 3.161613946654456, 3.617559217662347, 4.221297961772802, 4.680320341053220, 4.995422159259445, 5.342500803879590, 6.142190051728950, 6.662082855633976, 7.080259040490634, 7.679826469475099, 7.977970595691556, 8.645918749100543, 9.156620307468419, 9.423414161808490, 9.800477148852323, 10.52568601492658, 10.96738689688702, 11.45845960004852, 11.74686563893708, 12.20084659583979, 12.69078652742848
