| L(s) = 1 | − 9.20i·5-s + 58.8·7-s + 203. i·11-s − 40.3·13-s − 411. i·17-s − 674.·19-s + 295. i·23-s + 540.·25-s + 1.30e3i·29-s − 1.27e3·31-s − 541. i·35-s − 2.49e3·37-s − 1.60e3i·41-s + 995.·43-s − 2.49e3i·47-s + ⋯ |
| L(s) = 1 | − 0.368i·5-s + 1.20·7-s + 1.67i·11-s − 0.238·13-s − 1.42i·17-s − 1.86·19-s + 0.559i·23-s + 0.864·25-s + 1.54i·29-s − 1.32·31-s − 0.442i·35-s − 1.82·37-s − 0.956i·41-s + 0.538·43-s − 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.07852237157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07852237157\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9.20iT - 625T^{2} \) |
| 7 | \( 1 - 58.8T + 2.40e3T^{2} \) |
| 11 | \( 1 - 203. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 40.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 411. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 674.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 295. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.49e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.60e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 995.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.49e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.06e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.42e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.08e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.58e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.11e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 8.67e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.62e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.85e3T + 8.85e7T^{2} \) |
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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
−10.05452866254591167837791795729, −9.014699428785586474848368934813, −8.487502263961347646112923774190, −7.22254190215944300046201834696, −6.99243359639547706913310840037, −5.15797747432508978399373850499, −4.97726406510272098814928580114, −3.87419893870795192386941477912, −2.26081459349850736979494036625, −1.56750393118772377711743348428,
0.01613953898094008645433322499, 1.42306497380069794843310825595, 2.50044915839433864273699599410, 3.75204432038625327621135213832, 4.61634561794238860235363602788, 5.82152806197927728225318363625, 6.38953536460415469172861724746, 7.63578074274164932255776525427, 8.484795146567043157842349308898, 8.775445584852072655464359247346
