| L(s) = 1 | + 2-s + 4-s + 2.85·5-s + 7-s + 8-s + 2.85·10-s − 4.61·11-s + 5.23·13-s + 14-s + 16-s − 2.47·17-s − 19-s + 2.85·20-s − 4.61·22-s + 5.70·23-s + 3.14·25-s + 5.23·26-s + 28-s + 8.85·29-s − 0.472·31-s + 32-s − 2.47·34-s + 2.85·35-s + 4.85·37-s − 38-s + 2.85·40-s − 5.32·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.27·5-s + 0.377·7-s + 0.353·8-s + 0.902·10-s − 1.39·11-s + 1.45·13-s + 0.267·14-s + 0.250·16-s − 0.599·17-s − 0.229·19-s + 0.638·20-s − 0.984·22-s + 1.19·23-s + 0.629·25-s + 1.02·26-s + 0.188·28-s + 1.64·29-s − 0.0847·31-s + 0.176·32-s − 0.423·34-s + 0.482·35-s + 0.798·37-s − 0.162·38-s + 0.451·40-s − 0.831·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.806913464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806913464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 - 4.85T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.735912427785060027200071700491, −8.394723335542688423890482660872, −7.19989190130707047104776269913, −6.45455863470615685828068501095, −5.71953682293020473232997294757, −5.14738887025707870184973032707, −4.29666128466038975278772245658, −3.04421093520902745252441482350, −2.32795393932379781245519563327, −1.24999027785414522844343316450,
1.24999027785414522844343316450, 2.32795393932379781245519563327, 3.04421093520902745252441482350, 4.29666128466038975278772245658, 5.14738887025707870184973032707, 5.71953682293020473232997294757, 6.45455863470615685828068501095, 7.19989190130707047104776269913, 8.394723335542688423890482660872, 8.735912427785060027200071700491
