L(s) = 1 | − 28.5·3-s + (5.81 + 55.5i)5-s + 92.3i·7-s + 571.·9-s − 541. i·11-s − 782.·13-s + (−165. − 1.58e3i)15-s + 1.55e3i·17-s − 484. i·19-s − 2.63e3i·21-s − 1.09e3i·23-s + (−3.05e3 + 646. i)25-s − 9.37e3·27-s − 597. i·29-s − 8.86e3·31-s + ⋯ |
L(s) = 1 | − 1.83·3-s + (0.104 + 0.994i)5-s + 0.712i·7-s + 2.35·9-s − 1.34i·11-s − 1.28·13-s + (−0.190 − 1.82i)15-s + 1.30i·17-s − 0.307i·19-s − 1.30i·21-s − 0.432i·23-s + (−0.978 + 0.206i)25-s − 2.47·27-s − 0.131i·29-s − 1.65·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5522652773\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5522652773\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5.81 - 55.5i)T \) |
good | 3 | \( 1 + 28.5T + 243T^{2} \) |
| 7 | \( 1 - 92.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 541. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 782.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 484. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 597. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 8.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.44e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.06e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.77e4iT - 8.58e9T^{2} \) |
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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
−10.73375564869658595346636787381, −10.38910195181500576580461006182, −9.070797612126215510609677831575, −7.57660886846478160965084453628, −6.58709294403100546014586470409, −5.87532291819778543270074133210, −5.16441622498499821581068897004, −3.66982535080899838350088346941, −2.06892080171511769766836051235, −0.33936761427805172310649067783,
0.58826763222732401854033349528, 1.77888798406202610632934701803, 4.25248327414960150274676876526, 4.93631951511687021297091912476, 5.60835751038447259600255157580, 7.11528366329183409407683993968, 7.39816113102960967591825365393, 9.394996014448973586727677332256, 9.937254957157555585156279299950, 10.89207372750616113123928398490