| L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.413 + 0.459i)3-s + (−0.809 − 0.587i)4-s + (1.78 − 3.09i)5-s + (0.309 + 0.535i)6-s + (−0.309 + 2.94i)7-s + (−0.809 + 0.587i)8-s + (0.273 + 2.60i)9-s + (−2.39 − 2.65i)10-s + (−2.16 + 0.965i)11-s + (0.604 − 0.128i)12-s + (−5.53 − 1.17i)13-s + (2.70 + 1.20i)14-s + (0.682 + 2.10i)15-s + (0.309 + 0.951i)16-s + (2.83 + 1.26i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.238 + 0.265i)3-s + (−0.404 − 0.293i)4-s + (0.799 − 1.38i)5-s + (0.126 + 0.218i)6-s + (−0.116 + 1.11i)7-s + (−0.286 + 0.207i)8-s + (0.0912 + 0.867i)9-s + (−0.756 − 0.839i)10-s + (−0.654 + 0.291i)11-s + (0.174 − 0.0370i)12-s + (−1.53 − 0.326i)13-s + (0.721 + 0.321i)14-s + (0.176 + 0.542i)15-s + (0.0772 + 0.237i)16-s + (0.687 + 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.848706 - 0.384291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.848706 - 0.384291i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (4.84 + 2.74i)T \) |
| good | 3 | \( 1 + (0.413 - 0.459i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (-1.78 + 3.09i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.309 - 2.94i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (2.16 - 0.965i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (5.53 + 1.17i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.83 - 1.26i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.92 + 1.04i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 0.823i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.07 + 3.29i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.0938 + 0.162i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.60 - 1.78i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (11.0 - 2.34i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (0.00456 + 0.0140i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.425 + 4.04i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.00 + 5.56i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + (1.40 - 2.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.911 + 8.67i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (7.19 - 3.20i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.18 - 2.75i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 11.4i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.34 - 0.976i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.76 - 4.18i)T + (29.9 + 92.2i)T^{2} \) |
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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
−14.80049455874779289460403092294, −13.36768432973709807741916675560, −12.64837904613793746237714028512, −11.76384356299265510587845974889, −10.07375667980508498196595412478, −9.440045585932722604894639025987, −8.035969737907108567400748932910, −5.38473088977579549892485904416, −5.05546304975654657425411313163, −2.29641160850963261264562225980,
3.27308719821372791226150532040, 5.44499344304181360403738367522, 6.90613850966011603735014023399, 7.35275923943755253568160755527, 9.614429870941736498595112619909, 10.41414720985825017064945705262, 11.91725298367593050678223773456, 13.30385387526165987366609180335, 14.26196728063706115995312733924, 14.82260755179035559038236383187
