| L(s) = 1 | + (−1.41 − i)3-s − 2.23·5-s + (−1.41 − 2.23i)7-s + (1.00 + 2.82i)9-s − 2.82i·11-s − 2.82·13-s + (3.16 + 2.23i)15-s − 4i·17-s − 6.32i·19-s + (−0.236 + 4.57i)21-s − 6.32·23-s + 5.00·25-s + (1.41 − 5.00i)27-s + 5.65i·29-s + 6.32i·31-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)3-s − 0.999·5-s + (−0.534 − 0.845i)7-s + (0.333 + 0.942i)9-s − 0.852i·11-s − 0.784·13-s + (0.816 + 0.577i)15-s − 0.970i·17-s − 1.45i·19-s + (−0.0515 + 0.998i)21-s − 1.31·23-s + 1.00·25-s + (0.272 − 0.962i)27-s + 1.05i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02337431475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02337431475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4.47iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4.47iT - 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−9.577545892135294950591963152172, −8.640468834752301416602414657066, −7.67883708939669061519248054925, −7.10476447398083662854434470529, −6.62244497133821458089196052840, −5.40605876748584296662246173102, −4.67728618385684279333550136736, −3.68501572266880822936778865183, −2.61932196197713247195347312369, −0.856516053484519909405234030264,
0.01348944501963307379158627196, 1.99074337841513132935628741962, 3.43487935312739711970479603958, 4.16319910010350869275935812880, 4.96389323458345145835674062841, 5.99722097175160696818651464537, 6.50608871062870135208233639249, 7.70111744867357262401604322025, 8.242208369515064274604099735250, 9.412848861236384726434785044557
