| L(s) = 1 | + i·2-s − 4-s − 2.44·5-s − i·8-s − 2.44i·10-s − 4.24i·11-s − 0.717i·13-s + 16-s − 2.44·17-s − 4.89i·19-s + 2.44·20-s + 4.24·22-s + 6i·23-s + 0.999·25-s + 0.717·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.09·5-s − 0.353i·8-s − 0.774i·10-s − 1.27i·11-s − 0.198i·13-s + 0.250·16-s − 0.594·17-s − 1.12i·19-s + 0.547·20-s + 0.904·22-s + 1.25i·23-s + 0.199·25-s + 0.140·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6336031289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6336031289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 0.717iT - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 1.75iT - 29T^{2} \) |
| 31 | \( 1 - 9.08iT - 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 14.4iT - 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 4.18iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 5.49iT - 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 - 3.16iT - 97T^{2} \) |
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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
−8.928737515230450971568946280004, −8.282068249244250945731497451785, −7.66666172554034912747954815708, −6.95966355722388045452383367486, −6.14702432244868915102426287867, −5.31818560401698622244649109637, −4.45849164022975878812030367028, −3.63880931653081089649976108886, −2.83078959449366713392229140092, −0.974296306875470715659204744582,
0.25908327105876428669170023508, 1.79222205244829719042190605170, 2.68535881997343551267743626001, 4.00243269800576322729836035344, 4.19422762646589139150766650826, 5.17767759794147524341159931461, 6.34306744170816537766240625883, 7.16087502938463608532203302139, 8.014582683106447552186947954706, 8.414486176313004542930674134874
