| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + (2.40 − 4.38i)5-s + (6.94 − 0.853i)7-s − 2.82i·8-s + (6.20 + 3.39i)10-s − 2.88·11-s + 13.8·13-s + (1.20 + 9.82i)14-s + 4.00·16-s − 24.1·17-s − 6.53i·19-s + (−4.80 + 8.76i)20-s − 4.07i·22-s − 28.8i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + (0.480 − 0.876i)5-s + (0.992 − 0.121i)7-s − 0.353i·8-s + (0.620 + 0.339i)10-s − 0.261·11-s + 1.06·13-s + (0.0861 + 0.701i)14-s + 0.250·16-s − 1.42·17-s − 0.343i·19-s + (−0.240 + 0.438i)20-s − 0.185i·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.964135538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964135538\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.40 + 4.38i)T \) |
| 7 | \( 1 + (-6.94 + 0.853i)T \) |
| good | 11 | \( 1 + 2.88T + 121T^{2} \) |
| 13 | \( 1 - 13.8T + 169T^{2} \) |
| 17 | \( 1 + 24.1T + 289T^{2} \) |
| 19 | \( 1 + 6.53iT - 361T^{2} \) |
| 23 | \( 1 + 28.8iT - 529T^{2} \) |
| 29 | \( 1 - 32.9T + 841T^{2} \) |
| 31 | \( 1 - 2.43iT - 961T^{2} \) |
| 37 | \( 1 + 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 21.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 78.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 44.1T + 9.40e3T^{2} \) |
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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
−10.38324358806225217032507373534, −8.944034484234923551346911080609, −8.744166760506955729594090425233, −7.82638995502437662248280854255, −6.67449499423418683742347427693, −5.80773858808351363760679726850, −4.80514024938760492225941722417, −4.18868469378726660911659085282, −2.22258439871166524056604637494, −0.791249282494837890643047256873,
1.44430111021122096390915468244, 2.47535037775703756528041767848, 3.64838873965508600454119537634, 4.79031130872946775124038411017, 5.85370521445249888224499916314, 6.82476582850598599813856206991, 8.012537002683576822470338705355, 8.754799679048318383499428806108, 9.776140437881515508513428663829, 10.62458120359449621120206768042
