L(s) = 1 | + 1.60·3-s − 2.47·5-s + 7-s − 0.437·9-s + 1.48·11-s + 2.55·13-s − 3.95·15-s + 4.00·17-s + 1.95·19-s + 1.60·21-s + 3.60·23-s + 1.11·25-s − 5.50·27-s + 3.95·29-s + 4.47·31-s + 2.37·33-s − 2.47·35-s + 8.98·37-s + 4.08·39-s + 41-s − 2.00·43-s + 1.08·45-s − 0.268·47-s + 49-s + 6.41·51-s + 2.47·53-s − 3.67·55-s + ⋯ |
L(s) = 1 | + 0.924·3-s − 1.10·5-s + 0.377·7-s − 0.145·9-s + 0.447·11-s + 0.708·13-s − 1.02·15-s + 0.971·17-s + 0.447·19-s + 0.349·21-s + 0.750·23-s + 0.223·25-s − 1.05·27-s + 0.735·29-s + 0.803·31-s + 0.413·33-s − 0.418·35-s + 1.47·37-s + 0.654·39-s + 0.156·41-s − 0.306·43-s + 0.161·45-s − 0.0391·47-s + 0.142·49-s + 0.898·51-s + 0.339·53-s − 0.495·55-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
\(L(1)\) |
\(\approx\) |
\(2.013978831\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013978831\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.60T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 - 1.95T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 3.95T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 + 0.268T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 + 2.80T + 97T^{2} \) |
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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
−9.602213682293854585480066344685, −8.827316430854740816132636204410, −8.070626362779836082966207347584, −7.71837733274645521650151555153, −6.61540322787208078460651448843, −5.49762248688234440399865871808, −4.32401347716269278867755662324, −3.54621717942632038166476300302, −2.72800026276516320875672360667, −1.10215998988156029801508138359,
1.10215998988156029801508138359, 2.72800026276516320875672360667, 3.54621717942632038166476300302, 4.32401347716269278867755662324, 5.49762248688234440399865871808, 6.61540322787208078460651448843, 7.71837733274645521650151555153, 8.070626362779836082966207347584, 8.827316430854740816132636204410, 9.602213682293854585480066344685