| L(s) = 1 | + (0.652 − 0.652i)2-s + 3.14i·4-s + (4.83 − 1.27i)5-s + (8.09 − 8.09i)7-s + (4.66 + 4.66i)8-s + (2.32 − 3.98i)10-s + 2.60·11-s + (−2.08 − 2.08i)13-s − 10.5i·14-s − 6.49·16-s + (−13.6 + 13.6i)17-s − 11.4i·19-s + (3.99 + 15.2i)20-s + (1.70 − 1.70i)22-s + (7.89 + 7.89i)23-s + ⋯ |
| L(s) = 1 | + (0.326 − 0.326i)2-s + 0.786i·4-s + (0.967 − 0.254i)5-s + (1.15 − 1.15i)7-s + (0.583 + 0.583i)8-s + (0.232 − 0.398i)10-s + 0.237·11-s + (−0.160 − 0.160i)13-s − 0.754i·14-s − 0.406·16-s + (−0.805 + 0.805i)17-s − 0.602i·19-s + (0.199 + 0.761i)20-s + (0.0774 − 0.0774i)22-s + (0.343 + 0.343i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.69952 - 0.403375i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.69952 - 0.403375i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-4.83 + 1.27i)T \) |
| good | 2 | \( 1 + (-0.652 + 0.652i)T - 4iT^{2} \) |
| 7 | \( 1 + (-8.09 + 8.09i)T - 49iT^{2} \) |
| 11 | \( 1 - 2.60T + 121T^{2} \) |
| 13 | \( 1 + (2.08 + 2.08i)T + 169iT^{2} \) |
| 17 | \( 1 + (13.6 - 13.6i)T - 289iT^{2} \) |
| 19 | \( 1 + 11.4iT - 361T^{2} \) |
| 23 | \( 1 + (-7.89 - 7.89i)T + 529iT^{2} \) |
| 29 | \( 1 + 26.5iT - 841T^{2} \) |
| 31 | \( 1 - 43.7T + 961T^{2} \) |
| 37 | \( 1 + (-14.4 + 14.4i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 1.84T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-35.9 - 35.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (44.8 - 44.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-6.43 - 6.43i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 57.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 33.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (22.9 - 22.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 63.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (50.9 + 50.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 68.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (74.5 + 74.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (25.8 - 25.8i)T - 9.40e3iT^{2} \) |
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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
−11.04751253085095456844759851475, −10.33363109731263497294925215571, −9.126196881316583498106406675689, −8.182286731067651278862891341576, −7.38988556705260103668852352396, −6.22499167678456328697567945600, −4.75479292894106357415203751104, −4.24013771183338890987560134349, −2.64460653632616627193093762664, −1.37599633209726248094207377649,
1.50644303311655297926675010220, 2.53338601889853475737421206577, 4.64239197881359452841625666838, 5.30661080729638748473482800843, 6.17116816303893349813351769856, 7.04553888659180798430099706525, 8.490452630708618572840013640246, 9.267681389383253781833083401928, 10.15631564041716811355084945315, 11.07359792095811696697057798729
