| L(s) = 1 | − 3.88·5-s + (2.62 − 0.339i)7-s − 1.48·11-s + 2·13-s + 3.76i·17-s + 0.679i·19-s − 5.20i·23-s + 10.0·25-s + 7.03i·29-s − 6.42·31-s + (−10.1 + 1.32i)35-s − 8.71i·37-s − 3.16i·41-s − 8.20·43-s + 9.88·47-s + ⋯ |
| L(s) = 1 | − 1.73·5-s + (0.991 − 0.128i)7-s − 0.446·11-s + 0.554·13-s + 0.913i·17-s + 0.155i·19-s − 1.08i·23-s + 2.01·25-s + 1.30i·29-s − 1.15·31-s + (−1.72 + 0.223i)35-s − 1.43i·37-s − 0.493i·41-s − 1.25·43-s + 1.44·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7139643403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7139643403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.339i)T \) |
| good | 5 | \( 1 + 3.88T + 5T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.76iT - 17T^{2} \) |
| 19 | \( 1 - 0.679iT - 19T^{2} \) |
| 23 | \( 1 + 5.20iT - 23T^{2} \) |
| 29 | \( 1 - 7.03iT - 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + 8.71iT - 37T^{2} \) |
| 41 | \( 1 + 3.16iT - 41T^{2} \) |
| 43 | \( 1 + 8.20T + 43T^{2} \) |
| 47 | \( 1 - 9.88T + 47T^{2} \) |
| 53 | \( 1 + 6.42iT - 53T^{2} \) |
| 59 | \( 1 - 7.76iT - 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 8.81T + 67T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 4.44iT - 79T^{2} \) |
| 83 | \( 1 - 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 9.88iT - 97T^{2} \) |
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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
−8.558297567428485891902273773447, −7.931653412889356054164953765237, −7.40627787191124486928794754007, −6.68583958516277879491271239588, −5.53722788089917803456227714201, −4.83153893068856587989572243410, −3.94811958692293021224285979296, −3.59521761066029935272673248328, −2.27953161796830063214484788464, −1.05194115337305696251236064439,
0.24594965859372605432813499733, 1.50502126150712678040164428497, 2.84278574124399048067981246442, 3.62883057054776796958481335413, 4.45286612658444236685934761393, 5.00065610878957687063820178336, 5.92807987965008552845067780536, 7.07446725541115516040152731443, 7.60776818395033584073224435893, 8.081993630367176917497905264182
