| L(s) = 1 | + i·2-s + i·3-s − 4-s + (−1 − 2i)5-s − 6-s − i·8-s − 9-s + (2 − i)10-s − 4·11-s − i·12-s − 2i·13-s + (2 − i)15-s + 16-s − 6i·17-s − i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s + (0.516 − 0.258i)15-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.571051 - 0.352928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571051 - 0.352928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.46462821165226532758204302613, −9.517752060107356672020385457221, −8.766363184929676765386913323261, −7.930333423499553177034582419579, −7.20021440091719279360429174353, −5.68731223812146267683738564969, −5.07455768381776820349955560035, −4.22696197960079737462113218221, −2.83488330187070679192426662326, −0.37345075326660318719524411743,
1.79229596142358938527930914928, 2.96184052688672467759281624440, 3.93427986242902300787553498051, 5.34485935136836988743198872688, 6.40534708988336544380642055374, 7.49629294753819375148836117429, 8.106015319848450357798688585191, 9.210128340046148514239332393854, 10.42780428767633370412130043710, 10.76378627726636236112099785040
