L(s) = 1 | + 1.82i·3-s + 0.284i·5-s + 0.645i·7-s − 0.325·9-s − 1.56i·11-s + 1.12·13-s − 0.519·15-s + (1.35 + 3.89i)17-s − 0.845·19-s − 1.17·21-s − 2.49i·23-s + 4.91·25-s + 4.87i·27-s − 9.28i·29-s + 5.94i·31-s + ⋯ |
L(s) = 1 | + 1.05i·3-s + 0.127i·5-s + 0.243i·7-s − 0.108·9-s − 0.471i·11-s + 0.311·13-s − 0.134·15-s + (0.328 + 0.944i)17-s − 0.194·19-s − 0.256·21-s − 0.520i·23-s + 0.983·25-s + 0.938i·27-s − 1.72i·29-s + 1.06i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.932179730\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.932179730\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-1.35 - 3.89i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 1.82iT - 3T^{2} \) |
| 5 | \( 1 - 0.284iT - 5T^{2} \) |
| 7 | \( 1 - 0.645iT - 7T^{2} \) |
| 11 | \( 1 + 1.56iT - 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 19 | \( 1 + 0.845T + 19T^{2} \) |
| 23 | \( 1 + 2.49iT - 23T^{2} \) |
| 29 | \( 1 + 9.28iT - 29T^{2} \) |
| 31 | \( 1 - 5.94iT - 31T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 - 1.30iT - 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 0.319T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 61 | \( 1 + 4.55iT - 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 1.65iT - 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 6.36iT - 97T^{2} \) |
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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
−8.550659569592722749045137750584, −8.294906986724740881854569534695, −7.14442888062386319649938861433, −6.35255261475184954828629331275, −5.65239024144803044256649208712, −4.80948848428683243888171832467, −4.09955980250218002276346314963, −3.40429028471752051111934343132, −2.47214258248942519052754762062, −1.12739397734897435808662694310,
0.64239784104294278896617014653, 1.56100535121083354626224399756, 2.48318530663100274397172057898, 3.50587692404038614488464107907, 4.48837103828015358241101968639, 5.29492950015672042423090526281, 6.15279881962133913605117724583, 6.99350135769725769510219087461, 7.35405831774896253886856812560, 8.004075275936751887531081044029