| L(s) = 1 | − 1.34·3-s − 3.06·5-s + 7-s − 1.18·9-s + 4.82·11-s + 4.29·13-s + 4.12·15-s − 0.879·17-s − 5.57·19-s − 1.34·21-s + 5.10·23-s + 4.38·25-s + 5.63·27-s − 4.69·29-s − 10.1·31-s − 6.49·33-s − 3.06·35-s − 7.83·37-s − 5.78·39-s − 41-s − 0.509·43-s + 3.63·45-s − 4.61·47-s + 49-s + 1.18·51-s + 1.54·53-s − 14.7·55-s + ⋯ |
| L(s) = 1 | − 0.777·3-s − 1.37·5-s + 0.377·7-s − 0.394·9-s + 1.45·11-s + 1.19·13-s + 1.06·15-s − 0.213·17-s − 1.27·19-s − 0.294·21-s + 1.06·23-s + 0.877·25-s + 1.08·27-s − 0.871·29-s − 1.81·31-s − 1.13·33-s − 0.517·35-s − 1.28·37-s − 0.925·39-s − 0.156·41-s − 0.0777·43-s + 0.541·45-s − 0.673·47-s + 0.142·49-s + 0.165·51-s + 0.212·53-s − 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 0.879T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 43 | \( 1 + 0.509T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 1.54T + 53T^{2} \) |
| 59 | \( 1 + 8.12T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 7.38T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−8.964350210546061509597372679166, −8.791175552616402134537143950657, −7.71162638798377193368844624642, −6.79489000830137449689092706478, −6.12310154111000966163257743290, −5.03751918172783230612486832040, −4.04858607853162622169706611884, −3.43982480664638464943657582431, −1.50976129270845389347121498694, 0,
1.50976129270845389347121498694, 3.43982480664638464943657582431, 4.04858607853162622169706611884, 5.03751918172783230612486832040, 6.12310154111000966163257743290, 6.79489000830137449689092706478, 7.71162638798377193368844624642, 8.791175552616402134537143950657, 8.964350210546061509597372679166
