L(s) = 1 | + (2 + 2i)2-s + (11.7 − 11.7i)3-s + 8i·4-s + (−24.7 − 3.42i)5-s + 47.0·6-s + (−22.8 − 22.8i)7-s + (−16 + 16i)8-s − 195. i·9-s + (−42.6 − 56.3i)10-s − 90.9·11-s + (94.1 + 94.1i)12-s + (−139. + 139. i)13-s − 91.4i·14-s + (−331. + 250. i)15-s − 64·16-s + (−293. − 293. i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (1.30 − 1.30i)3-s + 0.5i·4-s + (−0.990 − 0.137i)5-s + 1.30·6-s + (−0.466 − 0.466i)7-s + (−0.250 + 0.250i)8-s − 2.41i·9-s + (−0.426 − 0.563i)10-s − 0.751·11-s + (0.653 + 0.653i)12-s + (−0.827 + 0.827i)13-s − 0.466i·14-s + (−1.47 + 1.11i)15-s − 0.250·16-s + (−1.01 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.357522762\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357522762\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 2i)T \) |
| 5 | \( 1 + (24.7 + 3.42i)T \) |
| 23 | \( 1 + (-77.9 + 77.9i)T \) |
good | 3 | \( 1 + (-11.7 + 11.7i)T - 81iT^{2} \) |
| 7 | \( 1 + (22.8 + 22.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 90.9T + 1.46e4T^{2} \) |
| 13 | \( 1 + (139. - 139. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (293. + 293. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 533. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 1.36e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.64e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.07e3 + 1.07e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.77e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.38e3 - 1.38e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.77e3 - 1.77e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-3.87e3 + 3.87e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.93e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (1.95e3 + 1.95e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 1.40e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.37e3 - 1.37e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 9.22e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (567. - 567. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 62.0iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.61e3 - 5.61e3i)T + 8.85e7iT^{2} \) |
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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
−11.74391349689188853313599152571, −9.877359227301430974933520522507, −8.676464630549820782600402445222, −7.894145589781361391838693813929, −7.22935956757772785334719768456, −6.46610040857010256085538694062, −4.52124442128642110173623087116, −3.37889825755747791528210436570, −2.26096417823701369113789913798, −0.30062739144029735647435588543,
2.62592563809827931583563519248, 3.15452358761859206753744807639, 4.37527776348569262526018944431, 5.14037911652989342265353918217, 7.11950002972372765371158876505, 8.428145331458668345253348724540, 8.971039458928589094949683665130, 10.33919828683597735304832592463, 10.60760214649578471268671698351, 11.95958672812638638458820787307