| L(s) = 1 | + 3.50i·5-s + 7-s + 3.01i·11-s + 3.90i·13-s + 1.38·17-s − 4.79i·19-s + 5.06·23-s − 7.25·25-s + 4.91i·29-s − 1.13·31-s + 3.50i·35-s + 9.45i·37-s + 4.11·41-s + 1.51i·43-s + 10.7·47-s + ⋯ |
| L(s) = 1 | + 1.56i·5-s + 0.377·7-s + 0.908i·11-s + 1.08i·13-s + 0.336·17-s − 1.09i·19-s + 1.05·23-s − 1.45·25-s + 0.911i·29-s − 0.204·31-s + 0.591i·35-s + 1.55i·37-s + 0.642·41-s + 0.231i·43-s + 1.57·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939471868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939471868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.50iT - 5T^{2} \) |
| 11 | \( 1 - 3.01iT - 11T^{2} \) |
| 13 | \( 1 - 3.90iT - 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 + 4.79iT - 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 4.91iT - 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 - 9.45iT - 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 - 1.51iT - 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.431iT - 53T^{2} \) |
| 59 | \( 1 + 7.40iT - 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 3.36iT - 67T^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 0.818T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.266501453236595982268475350327, −7.34120527676348929637540421458, −6.94321017238052101729412143438, −6.57200902529372915718819516506, −5.48582320777036646486076633230, −4.69050354640953909102381905893, −3.95285461611732286830926113178, −2.94534034144632051361694737611, −2.41667301605796098525325764047, −1.35416911155772787478846040610,
0.55030907290514497197411751403, 1.17701477855712497661007101952, 2.33166267519215332653749031767, 3.46938465717785764299637352667, 4.15551813386716660101334561895, 5.04562023087597780335091326478, 5.62167550200047941206602455885, 6.01626066622423739173224513025, 7.38064468718697120884040923020, 7.906160650127367072163979497998
