| L(s) = 1 | + (−1.32 − 1.80i)5-s + i·7-s − 2.48·11-s − 4.15i·13-s + 5.76i·17-s + 1.60·19-s − 7.28i·23-s + (−1.51 + 4.76i)25-s + 1.45·29-s − 2.24·31-s + (1.80 − 1.32i)35-s + 6i·37-s − 11.2·41-s − 5.28i·43-s + 3.45i·47-s + ⋯ |
| L(s) = 1 | + (−0.590 − 0.807i)5-s + 0.377i·7-s − 0.749·11-s − 1.15i·13-s + 1.39i·17-s + 0.369·19-s − 1.51i·23-s + (−0.303 + 0.952i)25-s + 0.270·29-s − 0.404·31-s + (0.305 − 0.223i)35-s + 0.986i·37-s − 1.76·41-s − 0.805i·43-s + 0.503i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3298576360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3298576360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.32 + 1.80i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 + 7.28iT - 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.28iT - 43T^{2} \) |
| 47 | \( 1 - 3.45iT - 47T^{2} \) |
| 53 | \( 1 - 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.52iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 7.28iT - 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.73iT - 97T^{2} \) |
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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
−8.897293298471173502900242323856, −8.338707343852302555290565357833, −7.957681013899598350533164758404, −6.94572518461306347545729417735, −5.91260491716056674272459690736, −5.26749221670850524785313010134, −4.48972827846643921426873194349, −3.52523163992515637473555000619, −2.57674070095778207485601798572, −1.23616658409360555570265336177,
0.11401191124837913187552011752, 1.81732444337859393287451200494, 2.96440028540264394697127446812, 3.66231536049378816869838850409, 4.65858354855003893101126239931, 5.43888733996543168059939791893, 6.56073240447391171011035319895, 7.23156705299691680905703718289, 7.62336933741006483103263908720, 8.598137893553393304077211697005
