| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 26.7·5-s + 51.4·7-s − 22.6i·8-s − 75.5i·10-s + 25.0·11-s − 53.2i·13-s + 145. i·14-s + 64.0·16-s + 24.4·17-s + (−78.0 + 352. i)19-s + 213.·20-s + 70.9i·22-s + 612.·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.06·5-s + 1.04·7-s − 0.353i·8-s − 0.755i·10-s + 0.207·11-s − 0.315i·13-s + 0.742i·14-s + 0.250·16-s + 0.0847·17-s + (−0.216 + 0.976i)19-s + 0.534·20-s + 0.146i·22-s + 1.15·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8976479915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8976479915\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (78.0 - 352. i)T \) |
| good | 5 | \( 1 + 26.7T + 625T^{2} \) |
| 7 | \( 1 - 51.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 25.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 53.2iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 24.4T + 8.35e4T^{2} \) |
| 23 | \( 1 - 612.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.34e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 912. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 325. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 51.7iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.54e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.99e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 4.06e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.07e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.53e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.38e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.80e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.62e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.90e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 997.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.15e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.02e3iT - 8.85e7T^{2} \) |
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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
−11.35977187281176694724203312632, −10.50045340101451223163813910169, −9.180923538319786777598390758131, −8.184892610540386852481092478407, −7.72903607467835718506491025884, −6.67451156764738482974228233641, −5.34962326987290931042885368799, −4.46887358905353126001513638915, −3.36485011581419257762312239246, −1.33869374863725754631994652274,
0.28898870728559711450147541191, 1.69326940337385670972221626952, 3.16238184890857202105181457884, 4.33863789148660415200738568156, 5.05172133468867353999806576680, 6.73967613798448917338676630906, 7.86091593233893095415288114782, 8.541181490031506070527429486350, 9.565725142560374721736364148646, 10.80813294053581186042533197265
