| L(s) = 1 | + (0.277 + 0.670i)2-s + (−0.962 − 1.44i)3-s + (2.45 − 2.45i)4-s + (1.19 − 6.01i)5-s + (0.698 − 1.04i)6-s + (2.29 + 11.5i)7-s + (5.01 + 2.07i)8-s + (−1.14 + 2.77i)9-s + (4.36 − 0.869i)10-s + (−9.91 − 6.62i)11-s + (−5.89 − 1.17i)12-s + (−2.41 − 2.41i)13-s + (−7.09 + 4.73i)14-s + (−9.81 + 4.06i)15-s − 9.95i·16-s + (−1.49 + 16.9i)17-s + ⋯ |
| L(s) = 1 | + (0.138 + 0.335i)2-s + (−0.320 − 0.480i)3-s + (0.613 − 0.613i)4-s + (0.239 − 1.20i)5-s + (0.116 − 0.174i)6-s + (0.327 + 1.64i)7-s + (0.626 + 0.259i)8-s + (−0.127 + 0.307i)9-s + (0.436 − 0.0869i)10-s + (−0.901 − 0.602i)11-s + (−0.491 − 0.0977i)12-s + (−0.185 − 0.185i)13-s + (−0.506 + 0.338i)14-s + (−0.654 + 0.271i)15-s − 0.621i·16-s + (−0.0882 + 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24409 - 0.284224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24409 - 0.284224i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.962 + 1.44i)T \) |
| 17 | \( 1 + (1.49 - 16.9i)T \) |
| good | 2 | \( 1 + (-0.277 - 0.670i)T + (-2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (-1.19 + 6.01i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-2.29 - 11.5i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (9.91 + 6.62i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (2.41 + 2.41i)T + 169iT^{2} \) |
| 19 | \( 1 + (-9.92 - 23.9i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (7.39 - 11.0i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (13.9 + 2.78i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (13.5 - 9.04i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (19.6 + 29.3i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (8.32 + 41.8i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-30.9 + 74.6i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-55.2 - 55.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (11.4 + 27.6i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-2.27 - 0.940i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (36.5 - 7.27i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 4.40iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (16.3 + 24.5i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-9.41 + 47.3i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 9.19i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (128. - 53.2i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (4.21 - 4.21i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-108. - 21.6i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−15.41964738938250329560008013189, −14.11598135150760567424374469811, −12.70903490619407598938203361771, −11.99288632179942076055757746876, −10.61224667619144086754349079075, −8.958843159647941682038745151541, −7.82499122044916399210523680670, −5.66931550081437441231139147493, −5.52388035141119296737675033157, −1.90137320823998971975383765631,
2.88818346982209978263572405618, 4.51458437720809198352512864980, 6.88084859660952108024546650392, 7.51598536646531790977651764807, 9.968264963774170276082133505773, 10.78911726665121641713017101176, 11.47979041874196914292271284076, 13.14382435604417416048508843740, 14.14453099451887416264861385884, 15.40798827496287262468378361973
