| L(s) = 1 | + (3.88 − 1.04i)2-s + (0.403 + 5.18i)3-s + (7.10 − 4.10i)4-s + (−0.168 − 0.168i)5-s + (6.96 + 19.7i)6-s + (5.26 − 19.6i)7-s + (0.579 − 0.579i)8-s + (−26.6 + 4.17i)9-s + (−0.832 − 0.480i)10-s + (−3.80 − 14.2i)11-s + (24.1 + 35.1i)12-s + (−27.1 − 38.2i)13-s − 81.8i·14-s + (0.806 − 0.942i)15-s + (−31.1 + 53.9i)16-s + (40.6 + 70.3i)17-s + ⋯ |
| L(s) = 1 | + (1.37 − 0.368i)2-s + (0.0775 + 0.996i)3-s + (0.888 − 0.512i)4-s + (−0.0151 − 0.0151i)5-s + (0.473 + 1.34i)6-s + (0.284 − 1.06i)7-s + (0.0255 − 0.0255i)8-s + (−0.987 + 0.154i)9-s + (−0.0263 − 0.0152i)10-s + (−0.104 − 0.389i)11-s + (0.580 + 0.845i)12-s + (−0.578 − 0.815i)13-s − 1.56i·14-s + (0.0138 − 0.0162i)15-s + (−0.486 + 0.843i)16-s + (0.579 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.34255 + 0.155442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34255 + 0.155442i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.403 - 5.18i)T \) |
| 13 | \( 1 + (27.1 + 38.2i)T \) |
| good | 2 | \( 1 + (-3.88 + 1.04i)T + (6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (0.168 + 0.168i)T + 125iT^{2} \) |
| 7 | \( 1 + (-5.26 + 19.6i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (3.80 + 14.2i)T + (-1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (-40.6 - 70.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-41.2 - 11.0i)T + (5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (65.0 - 112. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-138. - 79.7i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (9.94 - 9.94i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-315. + 84.5i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-53.2 + 14.2i)T + (5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (434. - 250. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-325. + 325. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 158. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (809. + 216. i)T + (1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (282. + 489. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-69.0 - 257. i)T + (-2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (169. - 634. i)T + (-3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (219. + 219. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-462. - 462. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (124. + 463. i)T + (-6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-92.2 - 24.7i)T + (7.90e5 + 4.56e5i)T^{2} \) |
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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
−15.38715939275614315475164657213, −14.40751630282880676092229361262, −13.65160471944606271178759008599, −12.27172574693738542616142520100, −11.00366461735117539723216224481, −10.04646712273009250824327944237, −8.078803175950804138600433493599, −5.78564769040016792748665113418, −4.48589790171104915404173618883, −3.28349376447135073422451232483,
2.63825840153561218969260068550, 4.95179589387467480865721365344, 6.23002702926493746043068260279, 7.51961632148090276671316891742, 9.247423859841296898474915531418, 11.81008410587077878123545458782, 12.19446569623981051236222187619, 13.48761238975500303403686062734, 14.37640763798236911909656952688, 15.25017700761450085912514901977
