| L(s) = 1 | + 3.09·3-s + 3.39·5-s − 1.58·7-s + 6.58·9-s + 5.36·11-s + 2.82·13-s + 10.5·15-s − 5.96·17-s + 19-s − 4.91·21-s + 5.19·23-s + 6.53·25-s + 11.1·27-s + 2.94·29-s − 7.59·31-s + 16.6·33-s − 5.38·35-s − 8.90·37-s + 8.73·39-s − 1.54·41-s + 8.09·43-s + 22.3·45-s − 5.47·47-s − 4.48·49-s − 18.4·51-s − 1.64·53-s + 18.2·55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s + 1.51·5-s − 0.599·7-s + 2.19·9-s + 1.61·11-s + 0.782·13-s + 2.71·15-s − 1.44·17-s + 0.229·19-s − 1.07·21-s + 1.08·23-s + 1.30·25-s + 2.13·27-s + 0.547·29-s − 1.36·31-s + 2.89·33-s − 0.910·35-s − 1.46·37-s + 1.39·39-s − 0.242·41-s + 1.23·43-s + 3.33·45-s − 0.798·47-s − 0.640·49-s − 2.58·51-s − 0.225·53-s + 2.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.821539839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.821539839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 + 7.59T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.982T + 73T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 0.628T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.418285831854666106551514631373, −7.29006897591734179239343569072, −6.60914475881512606211138860517, −6.31972349678633179366666104751, −5.14423396882900777689239888712, −4.15919808770313353090250785702, −3.47491862753856688651028103286, −2.78626771090448138348698700304, −1.86012745008580213669886927625, −1.38990612371960997964690448029,
1.38990612371960997964690448029, 1.86012745008580213669886927625, 2.78626771090448138348698700304, 3.47491862753856688651028103286, 4.15919808770313353090250785702, 5.14423396882900777689239888712, 6.31972349678633179366666104751, 6.60914475881512606211138860517, 7.29006897591734179239343569072, 8.418285831854666106551514631373
