| L(s) = 1 | + 3i·3-s + (−10 − 5i)5-s − 10i·7-s − 9·9-s + 46·11-s + 34i·13-s + (15 − 30i)15-s + 66i·17-s + 104·19-s + 30·21-s + 164i·23-s + (75 + 100i)25-s − 27i·27-s − 224·29-s + 72·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.539i·7-s − 0.333·9-s + 1.26·11-s + 0.725i·13-s + (0.258 − 0.516i)15-s + 0.941i·17-s + 1.25·19-s + 0.311·21-s + 1.48i·23-s + (0.599 + 0.800i)25-s − 0.192i·27-s − 1.43·29-s + 0.417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.23325 + 0.762193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23325 + 0.762193i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (10 + 5i)T \) |
| good | 7 | \( 1 + 10iT - 343T^{2} \) |
| 11 | \( 1 - 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 104T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 224T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 194T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 426T + 2.05e5T^{2} \) |
| 61 | \( 1 - 698T + 2.26e5T^{2} \) |
| 67 | \( 1 + 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 188T + 3.57e5T^{2} \) |
| 73 | \( 1 - 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−11.56443050906327530517220662032, −11.21397825538864847938256684229, −9.727043493866489091855245491148, −9.104298777559835004724024263880, −7.922391735247120713232863085489, −6.96437411932091214153296068620, −5.54211626932459469915570398594, −4.17196759682810013289975334958, −3.62648118495801271238709323632, −1.25387175509316289668548395249,
0.70148799396917029260058376361, 2.62346782158293543362061476002, 3.84986752814880570112904772219, 5.38159035300773826298257328281, 6.65502269083394473260254694649, 7.44222762645074758291071728648, 8.475669382080289593855659949869, 9.446368007447710611995930170548, 10.77716432500795205941830858081, 11.82743919665936893472680894648
