| L(s) = 1 | + i·2-s − 4-s + 2.74i·5-s + (−2.24 − 1.39i)7-s − i·8-s − 2.74·10-s + 1.52·11-s + 5.33·13-s + (1.39 − 2.24i)14-s + 16-s + 2.78i·17-s + (−4.01 − 1.68i)19-s − 2.74i·20-s + 1.52i·22-s − 6.49·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.22i·5-s + (−0.849 − 0.527i)7-s − 0.353i·8-s − 0.867·10-s + 0.461·11-s + 1.48·13-s + (0.372 − 0.600i)14-s + 0.250·16-s + 0.676i·17-s + (−0.921 − 0.387i)19-s − 0.613i·20-s + 0.326i·22-s − 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6545093872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6545093872\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.24 + 1.39i)T \) |
| 19 | \( 1 + (4.01 + 1.68i)T \) |
| good | 5 | \( 1 - 2.74iT - 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 - 2.78iT - 17T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 6.88iT - 29T^{2} \) |
| 31 | \( 1 + 0.830T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.668T + 43T^{2} \) |
| 47 | \( 1 - 4.83iT - 47T^{2} \) |
| 53 | \( 1 - 2.02iT - 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 + 7.03iT - 61T^{2} \) |
| 67 | \( 1 + 3.43iT - 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 4.32iT - 79T^{2} \) |
| 83 | \( 1 + 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−9.310015959743146927493942366931, −8.548428380085395605009842639267, −7.82577988518601812411939067628, −6.79793883115659220856833445057, −6.45695225676967588349281434367, −6.00327269814813935175090472754, −4.59906497498966438002473625202, −3.63132315417473706928062544116, −3.20112829086452356187954505329, −1.59750569909296264189942503255,
0.22530236566917116635404377149, 1.45014489870084292666163518882, 2.46801966034558626333865274996, 3.80939287817801803495183266645, 4.12482578687969898030039491315, 5.41038148187591540766615730700, 5.93879979667982550694448424621, 6.83990553782372953325488242134, 8.277395375922016069852296741119, 8.524475584588177260563350502891
