| L(s) = 1 | + 0.00586·3-s + 2.78·5-s − 4.08·7-s − 2.99·9-s − 0.755·11-s − 2.25·13-s + 0.0163·15-s − 2.34·17-s + 19-s − 0.0239·21-s + 5.02·23-s + 2.73·25-s − 0.0351·27-s − 4.98·29-s + 2.57·31-s − 0.00442·33-s − 11.3·35-s + 9.55·37-s − 0.0131·39-s − 1.74·41-s + 11.3·43-s − 8.34·45-s + 2.44·47-s + 9.66·49-s − 0.0137·51-s + 53-s − 2.10·55-s + ⋯ |
| L(s) = 1 | + 0.00338·3-s + 1.24·5-s − 1.54·7-s − 0.999·9-s − 0.227·11-s − 0.624·13-s + 0.00421·15-s − 0.568·17-s + 0.229·19-s − 0.00522·21-s + 1.04·23-s + 0.547·25-s − 0.00676·27-s − 0.925·29-s + 0.462·31-s − 0.000770·33-s − 1.91·35-s + 1.57·37-s − 0.00211·39-s − 0.273·41-s + 1.73·43-s − 1.24·45-s + 0.356·47-s + 1.38·49-s − 0.00192·51-s + 0.137·53-s − 0.283·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511053323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511053323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 0.00586T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 0.755T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 8.42T + 61T^{2} \) |
| 67 | \( 1 + 3.97T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 + 9.20T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−8.727508244231696716534474078391, −7.58219187024804687996921866579, −6.85233987491009675802376468649, −6.06352400587952234851345055328, −5.74446734505282265864910926111, −4.85074673002464218340332444128, −3.66364972278517049806837573063, −2.71563711014311786272090037467, −2.32699157166775921711145130594, −0.66779666644785765062736346519,
0.66779666644785765062736346519, 2.32699157166775921711145130594, 2.71563711014311786272090037467, 3.66364972278517049806837573063, 4.85074673002464218340332444128, 5.74446734505282265864910926111, 6.06352400587952234851345055328, 6.85233987491009675802376468649, 7.58219187024804687996921866579, 8.727508244231696716534474078391
