L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1 + i)7-s + (0.707 + 0.707i)8-s − 1.41i·11-s + (1 − i)13-s − 1.41·14-s − 1.00·16-s + (1.00 + 1.00i)22-s + 1.41i·26-s + (1.00 − 1.00i)28-s + (0.707 − 0.707i)32-s + (−1 − i)37-s − 1.41i·41-s − 1.41·44-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1 + i)7-s + (0.707 + 0.707i)8-s − 1.41i·11-s + (1 − i)13-s − 1.41·14-s − 1.00·16-s + (1.00 + 1.00i)22-s + 1.41i·26-s + (1.00 − 1.00i)28-s + (0.707 − 0.707i)32-s + (−1 − i)37-s − 1.41i·41-s − 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9262871019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9262871019\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 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
−9.092855305422953962239594127733, −8.623030719936432346329853296659, −8.174120784450892849727518871352, −7.34453457457955326215505815771, −6.13288875337869690260851175448, −5.69006235946435152858730382070, −5.03585486811106907809975034803, −3.63321752741537757675417626531, −2.35199168979366122086570932528, −1.07594996192474696148767366100,
1.36612398721585516636666245689, 2.04303132672494496543575639120, 3.55958993731141655793156347218, 4.30613868598206531911541445269, 4.99140682932471197329706964631, 6.72604773295394192365750713765, 7.07729504087234833628111914660, 8.093345337789969405866189888492, 8.553058277293172819439134398803, 9.610197079978510139822138056319