| L(s) = 1 | + 3.09·3-s + 4.37i·7-s + 6.58·9-s − 2.53i·11-s + (−2.44 + 2.64i)13-s − 1.29·17-s + 5.36i·19-s + 13.5i·21-s + 1.80·23-s + 11.0·27-s + 7.58·29-s − 3.12i·31-s − 7.84i·33-s − 2.55i·37-s + (−7.58 + 8.19i)39-s + ⋯ |
| L(s) = 1 | + 1.78·3-s + 1.65i·7-s + 2.19·9-s − 0.763i·11-s + (−0.679 + 0.733i)13-s − 0.313·17-s + 1.23i·19-s + 2.95i·21-s + 0.376·23-s + 2.13·27-s + 1.40·29-s − 0.561i·31-s − 1.36i·33-s − 0.419i·37-s + (−1.21 + 1.31i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.067184829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.067184829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.44 - 2.64i)T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 11 | \( 1 + 2.53iT - 11T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 5.36iT - 19T^{2} \) |
| 23 | \( 1 - 1.80T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.12iT - 31T^{2} \) |
| 37 | \( 1 + 2.55iT - 37T^{2} \) |
| 41 | \( 1 + 7.89iT - 41T^{2} \) |
| 43 | \( 1 + 4.38T + 43T^{2} \) |
| 47 | \( 1 - 6.20iT - 47T^{2} \) |
| 53 | \( 1 - 6.19T + 53T^{2} \) |
| 59 | \( 1 + 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 + 2.55iT - 67T^{2} \) |
| 71 | \( 1 + 0.295iT - 71T^{2} \) |
| 73 | \( 1 - 2.55iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.723iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−9.499883611812031639422556094275, −8.745793748932489448616014897565, −8.473475466828579056512500229313, −7.59853554220418131317359653285, −6.56988136216663675125831476954, −5.57578489900941597059782590256, −4.43255046372445491306373902472, −3.36920863492703560293956640180, −2.56767967674782442331308528989, −1.84644914110839402114065880590,
1.11287241477988992620201586610, 2.49341469082704368558895010841, 3.25463585168428906871724706870, 4.29384964598336035627183567388, 4.84213295759570747601206934486, 6.93582457559557935964458113512, 7.06028237449287823413742753565, 7.986954865675671670099166273287, 8.617512688087918847055865230105, 9.607585904675547607890931122032
