| L(s) = 1 | + (0.309 − 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (0.408 − 3.88i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (3.25 − 1.44i)11-s + (−0.978 + 0.207i)12-s + (−3.59 − 0.764i)13-s + (−3.56 − 1.58i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.319 − 0.142i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.386 − 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.154 − 1.46i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (0.211 + 0.235i)10-s + (0.980 − 0.436i)11-s + (−0.282 + 0.0600i)12-s + (−0.998 − 0.212i)13-s + (−0.953 − 0.424i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.0774 − 0.0344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.281567 - 1.54480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.281567 - 1.54480i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-4.71 - 2.96i)T \) |
| good | 7 | \( 1 + (-0.408 + 3.88i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.25 + 1.44i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.59 + 0.764i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.319 + 0.142i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (2.66 - 0.566i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.27 + 0.928i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.283 + 0.872i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (3.13 + 5.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.39 + 5.99i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (9.44 - 2.00i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (1.30 + 4.01i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.146 + 1.39i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.16 + 6.84i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 + (1.34 - 2.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.746 - 7.10i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.290 + 0.129i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-14.9 - 6.65i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.51 + 7.23i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.58 - 3.32i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.44 - 3.22i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.996145659601413677756984961791, −8.877297640522854822308154257024, −8.042698090507230659502403273494, −7.05741699066738693398211380219, −6.54424604373575543694440099559, −5.03567438680958847240021167660, −4.00626692582739164035086169125, −3.33739071346064971768848354139, −2.00900511437201687090552968646, −0.65568597197372363671241111734,
2.01634136575593176715223297548, 3.23720748465118876362026244720, 4.53054797745734228932872672894, 5.02304537768428812211746025149, 6.13594462361486447769533425767, 6.98226176517544753530870773791, 8.124195282153077242288877743512, 8.712382019702665775794739336039, 9.378069462591211239742154149506, 10.07042961636550673452157099550
