| L(s) = 1 | − 3.25·3-s − 2.40·5-s − 4.10·7-s + 7.59·9-s + 5.28·11-s − 1.26·13-s + 7.83·15-s − 2.46·17-s − 4.09·19-s + 13.3·21-s − 0.252·23-s + 0.790·25-s − 14.9·27-s − 2.53·29-s + 0.649·31-s − 17.2·33-s + 9.88·35-s + 6.06·37-s + 4.12·39-s − 9.02·41-s − 8.38·43-s − 18.2·45-s − 10.6·47-s + 9.86·49-s + 8.03·51-s + 5.73·53-s − 12.7·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s − 1.07·5-s − 1.55·7-s + 2.53·9-s + 1.59·11-s − 0.351·13-s + 2.02·15-s − 0.598·17-s − 0.940·19-s + 2.91·21-s − 0.0526·23-s + 0.158·25-s − 2.87·27-s − 0.471·29-s + 0.116·31-s − 2.99·33-s + 1.67·35-s + 0.997·37-s + 0.659·39-s − 1.41·41-s − 1.27·43-s − 2.72·45-s − 1.54·47-s + 1.40·49-s + 1.12·51-s + 0.787·53-s − 1.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1495061458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1495061458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 0.252T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 0.649T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.38T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 8.81T + 97T^{2} \) |
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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
−8.449445839073653590723808902358, −7.29560097566130605939163563535, −6.69281491441432682318954306408, −6.42990520057375151421418305910, −5.66301195126678678197662087753, −4.51726464368539443435609870977, −4.11942872862675741932200823854, −3.28458686640781384229883912925, −1.57048806082167829300374571704, −0.24756876820128438707298113999,
0.24756876820128438707298113999, 1.57048806082167829300374571704, 3.28458686640781384229883912925, 4.11942872862675741932200823854, 4.51726464368539443435609870977, 5.66301195126678678197662087753, 6.42990520057375151421418305910, 6.69281491441432682318954306408, 7.29560097566130605939163563535, 8.449445839073653590723808902358
