| L(s) = 1 | + 1.41i·5-s + 2·7-s + 1.41i·11-s + 6.32i·13-s + 4.24i·17-s + (3 − 3.16i)19-s − 7.07i·23-s + 2.99·25-s − 4.47·29-s + 6.32i·31-s + 2.82i·35-s + 4.47·41-s − 4·43-s + 7.07i·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s + 0.755·7-s + 0.426i·11-s + 1.75i·13-s + 1.02i·17-s + (0.688 − 0.725i)19-s − 1.47i·23-s + 0.599·25-s − 0.830·29-s + 1.13i·31-s + 0.478i·35-s + 0.698·41-s − 0.609·43-s + 1.03i·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757814997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757814997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3 + 3.16i)T \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 6.32iT - 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 6.32iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 1.41iT - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 6.32iT - 97T^{2} \) |
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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
−9.041504614022403503018149906709, −8.305152797254499624841506161923, −7.44034984242749822634541910045, −6.73071401213636546668287237367, −6.21833839051215522768527217794, −4.88977620412708082413290471036, −4.47942923083300307876074944368, −3.39305720260309779602237103757, −2.30105106626850360971507890753, −1.45092584749131381905973905005,
0.58761729278060226075713295127, 1.61939724843390129630547279697, 2.96168702141888287387910092955, 3.71451424182130498543812970706, 5.00220754565280265466849498750, 5.31209066896230370925768721154, 6.10381519300904640466639238464, 7.45301574159304812127339027561, 7.81885010598575828045199118167, 8.477093673357199245579443102344
