| L(s) = 1 | + (1.74 − 1.40i)5-s + 2i·7-s + 2.48·11-s − 2.80i·13-s − 4.80i·17-s − 6.48·19-s + 6.96i·23-s + (1.06 − 4.88i)25-s + 8.61·29-s + 8.09·31-s + (2.80 + 3.48i)35-s − 6.15i·37-s − 10.0·41-s − 6i·43-s − 8.96i·47-s + ⋯ |
| L(s) = 1 | + (0.778 − 0.627i)5-s + 0.755i·7-s + 0.748·11-s − 0.778i·13-s − 1.16i·17-s − 1.48·19-s + 1.45i·23-s + (0.212 − 0.977i)25-s + 1.59·29-s + 1.45·31-s + (0.474 + 0.588i)35-s − 1.01i·37-s − 1.57·41-s − 0.914i·43-s − 1.30i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.164037859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.164037859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.74 + 1.40i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 + 2.80iT - 13T^{2} \) |
| 17 | \( 1 + 4.80iT - 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 - 6.96iT - 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 + 6.15iT - 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8.96iT - 47T^{2} \) |
| 53 | \( 1 + 0.965iT - 53T^{2} \) |
| 59 | \( 1 - 0.871T + 59T^{2} \) |
| 61 | \( 1 - 3.48T + 61T^{2} \) |
| 67 | \( 1 + 6.96iT - 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 - 8.80iT - 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 0.965iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 4.96iT - 97T^{2} \) |
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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
−8.700965761490320661425721908082, −8.016051191501244203207993137780, −6.85603922891063312375071799245, −6.28994623234562498403463186243, −5.39878904315454285049326980318, −4.94095540976420912539284847061, −3.86476870106923043857742570619, −2.73985188470369057107333690781, −1.94310576580173370027553033226, −0.72717218722565202216607676382,
1.18088820942070033690940334625, 2.15792042148073725055242503183, 3.12410057016290395049179522548, 4.26642757523312404404808387740, 4.63384804682370106189972932217, 6.17852261814175831920619548891, 6.45929784800565682119949064846, 6.92754156345751339978995448910, 8.254689983357443376548711421015, 8.585426848600250985664953115123
