| L(s) = 1 | + (0.130 − 0.991i)2-s + (−0.980 − 0.195i)3-s + (−0.442 + 0.896i)5-s + (−0.321 + 0.946i)6-s + (−0.442 − 0.896i)7-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (0.831 + 0.555i)10-s + (0.946 + 0.321i)11-s + (−0.258 + 0.965i)13-s + (−0.946 + 0.321i)14-s + (0.608 − 0.793i)15-s + (−0.866 − 0.499i)16-s + (0.5 − 0.866i)18-s + (0.258 + 0.965i)21-s + (0.442 − 0.896i)22-s + ⋯ |
| L(s) = 1 | + (0.130 − 0.991i)2-s + (−0.980 − 0.195i)3-s + (−0.442 + 0.896i)5-s + (−0.321 + 0.946i)6-s + (−0.442 − 0.896i)7-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (0.831 + 0.555i)10-s + (0.946 + 0.321i)11-s + (−0.258 + 0.965i)13-s + (−0.946 + 0.321i)14-s + (0.608 − 0.793i)15-s + (−0.866 − 0.499i)16-s + (0.5 − 0.866i)18-s + (0.258 + 0.965i)21-s + (0.442 − 0.896i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.005418127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005418127\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.980 + 0.195i)T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.130 + 0.991i)T + (-0.965 - 0.258i)T^{2} \) |
| 5 | \( 1 + (0.442 - 0.896i)T + (-0.608 - 0.793i)T^{2} \) |
| 7 | \( 1 + (0.442 + 0.896i)T + (-0.608 + 0.793i)T^{2} \) |
| 11 | \( 1 + (-0.946 - 0.321i)T + (0.793 + 0.608i)T^{2} \) |
| 13 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (0.659 + 0.751i)T + (-0.130 + 0.991i)T^{2} \) |
| 29 | \( 1 + (-0.0654 - 0.997i)T + (-0.991 + 0.130i)T^{2} \) |
| 31 | \( 1 + (-0.946 + 0.321i)T + (0.793 - 0.608i)T^{2} \) |
| 37 | \( 1 + (-1.96 - 0.390i)T + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.997 - 0.0654i)T + (0.991 + 0.130i)T^{2} \) |
| 43 | \( 1 + (-0.793 + 0.608i)T + (0.258 - 0.965i)T^{2} \) |
| 47 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.991 - 0.130i)T + (0.965 - 0.258i)T^{2} \) |
| 61 | \( 1 + (-0.896 + 0.442i)T + (0.608 - 0.793i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (0.946 + 0.321i)T + (0.793 + 0.608i)T^{2} \) |
| 83 | \( 1 + (-0.991 - 0.130i)T + (0.965 + 0.258i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (0.0654 + 0.997i)T + (-0.991 + 0.130i)T^{2} \) |
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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
−9.392330079841902162135711944244, −7.87505727519752540505604129246, −7.04424433357925339008359637497, −6.76236195854255323214645014887, −6.11506876856395167946551685962, −4.44615020401828693527223881254, −4.19530042287495582826783920240, −3.20412649884682980237422023307, −2.10680084869838349603068020022, −0.971342348257267724086000990066,
1.02271782801025996347432358165, 2.57811344162070198985893761125, 4.02107710985602954354160917914, 4.73639018471373469105624339509, 5.62256789162324944315963428709, 6.00922478487272492199715133843, 6.63832077137042405960827934204, 7.73953181655498869236412766667, 8.167618475983500120720009196390, 9.186362017164865668209006228849
