L(s) = 1 | + (1.30 + 1.30i)2-s + 2.41i·4-s + (−0.923 + 0.382i)5-s + (−1.84 + 1.84i)8-s + (−1.70 − 0.707i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (−0.923 − 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s + 1.84i·26-s + (−1.30 − 1.30i)32-s + (1.00 − 2.41i)40-s + 0.765i·41-s + ⋯ |
L(s) = 1 | + (1.30 + 1.30i)2-s + 2.41i·4-s + (−0.923 + 0.382i)5-s + (−1.84 + 1.84i)8-s + (−1.70 − 0.707i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (−0.923 − 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s + 1.84i·26-s + (−1.30 − 1.30i)32-s + (1.00 − 2.41i)40-s + 0.765i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.496548709\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496548709\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.84iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 0.765iT - T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 0.765T + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 0.765iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 89 | \( 1 + 1.84T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−11.51698535593075887418137425205, −10.74190454206365760751838209808, −8.873055399219407593816508098266, −8.299254130005556637255294869504, −7.46539072737060528135118735333, −6.49936911497139519509939936504, −5.92832096469139046492801982925, −4.75563794454157060964909283617, −3.73894123633339661424945036085, −3.13614529021210741099343827663,
1.51692530949626808455739992371, 2.89638044487971571233836669821, 3.99095436727907082816390778127, 4.63467925979035227287303361122, 5.53254811583956369996033120175, 6.81751593913886566727291587568, 7.921363545392969964933473877695, 9.220004180320881461686229263664, 10.10957663507210970371288539767, 10.86695682967106755659582272211