L(s) = 1 | + i·5-s + (−1.81 − 1.04i)7-s + (−1.5 + 0.866i)11-s + (−3.59 + 0.331i)13-s + (1.81 − 3.14i)17-s + (0.926 + 0.534i)19-s + (3.90 + 6.77i)23-s − 25-s + (−0.263 − 0.456i)29-s − 5.84i·31-s + (1.04 − 1.81i)35-s + (8.44 − 4.87i)37-s + (3.69 − 2.13i)41-s + (4.67 − 8.09i)43-s − 3.46i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + (−0.685 − 0.395i)7-s + (−0.452 + 0.261i)11-s + (−0.995 + 0.0918i)13-s + (0.439 − 0.762i)17-s + (0.212 + 0.122i)19-s + (0.815 + 1.41i)23-s − 0.200·25-s + (−0.0489 − 0.0847i)29-s − 1.04i·31-s + (0.177 − 0.306i)35-s + (1.38 − 0.801i)37-s + (0.577 − 0.333i)41-s + (0.712 − 1.23i)43-s − 0.505i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.212653778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212653778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.59 - 0.331i)T \) |
good | 7 | \( 1 + (1.81 + 1.04i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.926 - 0.534i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.90 - 6.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.263 + 0.456i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.84iT - 31T^{2} \) |
| 37 | \( 1 + (-8.44 + 4.87i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.69 + 2.13i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.67 + 8.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + (-1.21 - 0.701i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.55 + 9.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.38 - 5.41i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 7.08i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2.64iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 15.7iT - 83T^{2} \) |
| 89 | \( 1 + (-4.78 + 2.76i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 + 7.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.153454436276782973011502780888, −7.66013953765258876737855019051, −7.53174202304609996305886636177, −6.66794699279615044805067553443, −5.70252886098175221092762178999, −4.97154159533787929789383674064, −3.88964091627014337845880700973, −3.04752457817462912955352447715, −2.16076627489012176300770402733, −0.48972243828317617337283311988,
1.00808187269343514342939983688, 2.52465724382750437700388593095, 3.15968203332135762830888872964, 4.45940854002470733954914748881, 5.07969513281439639502965040656, 6.05806657891574735566731378342, 6.65049070607598746037994673477, 7.75360338822753142583835941185, 8.257034539448475017545480283933, 9.254803042887495160232257825144