| L(s) = 1 | − 3.84·2-s + 26.7·3-s − 17.2·4-s + 25·5-s − 103.·6-s + 117.·7-s + 189.·8-s + 475.·9-s − 96.1·10-s + 121·11-s − 461.·12-s + 571.·13-s − 453.·14-s + 669.·15-s − 176.·16-s + 1.32e3·17-s − 1.82e3·18-s + 361·19-s − 430.·20-s + 3.15e3·21-s − 465.·22-s + 1.79e3·23-s + 5.07e3·24-s + 625·25-s − 2.19e3·26-s + 6.22e3·27-s − 2.02e3·28-s + ⋯ |
| L(s) = 1 | − 0.679·2-s + 1.71·3-s − 0.538·4-s + 0.447·5-s − 1.16·6-s + 0.909·7-s + 1.04·8-s + 1.95·9-s − 0.303·10-s + 0.301·11-s − 0.925·12-s + 0.937·13-s − 0.617·14-s + 0.768·15-s − 0.172·16-s + 1.11·17-s − 1.32·18-s + 0.229·19-s − 0.240·20-s + 1.56·21-s − 0.204·22-s + 0.706·23-s + 1.79·24-s + 0.200·25-s − 0.636·26-s + 1.64·27-s − 0.489·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.369185498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.369185498\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 3.84T + 32T^{2} \) |
| 3 | \( 1 - 26.7T + 243T^{2} \) |
| 7 | \( 1 - 117.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 571.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 882.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.58e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.28e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 694.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.62e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.98e3T + 8.58e9T^{2} \) |
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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
−8.976251934761381339985386703001, −8.561652113265795226804475965893, −7.82012852006562451978374437370, −7.24261484055006728472865251548, −5.74215579862094396515710122966, −4.59806903407048357551957420716, −3.79234623739861797258446798533, −2.79282474340376147589275429086, −1.57181930843155997131076474685, −1.08811075737375676974004054973,
1.08811075737375676974004054973, 1.57181930843155997131076474685, 2.79282474340376147589275429086, 3.79234623739861797258446798533, 4.59806903407048357551957420716, 5.74215579862094396515710122966, 7.24261484055006728472865251548, 7.82012852006562451978374437370, 8.561652113265795226804475965893, 8.976251934761381339985386703001
