| L(s) = 1 | + 4·2-s − 8.12·3-s + 16·4-s + 20.9·5-s − 32.4·6-s − 59.5·7-s + 64·8-s − 177.·9-s + 83.8·10-s + 652.·11-s − 129.·12-s − 914.·13-s − 238.·14-s − 170.·15-s + 256·16-s − 207.·17-s − 708.·18-s − 535.·19-s + 335.·20-s + 483.·21-s + 2.61e3·22-s + 3.52e3·23-s − 519.·24-s − 2.68e3·25-s − 3.65e3·26-s + 3.41e3·27-s − 952.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.520·3-s + 0.5·4-s + 0.374·5-s − 0.368·6-s − 0.459·7-s + 0.353·8-s − 0.728·9-s + 0.265·10-s + 1.62·11-s − 0.260·12-s − 1.50·13-s − 0.324·14-s − 0.195·15-s + 0.250·16-s − 0.174·17-s − 0.515·18-s − 0.340·19-s + 0.187·20-s + 0.239·21-s + 1.15·22-s + 1.39·23-s − 0.184·24-s − 0.859·25-s − 1.06·26-s + 0.900·27-s − 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.681743867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.681743867\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 269 | \( 1 + 7.23e4T \) |
| good | 3 | \( 1 + 8.12T + 243T^{2} \) |
| 5 | \( 1 - 20.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 59.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 652.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 914.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 207.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 535.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 24.8T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.66e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.63e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−10.00344709845432443027923473254, −9.374723244681234765771124099752, −8.219609508969567811431764434637, −6.77156821555178630858224275211, −6.47120282126823491157982802144, −5.34624962951574395749898329460, −4.52324614216963712690827272095, −3.28603183019328689792258938989, −2.22230683705874773779849238077, −0.74053517594024765282978609568,
0.74053517594024765282978609568, 2.22230683705874773779849238077, 3.28603183019328689792258938989, 4.52324614216963712690827272095, 5.34624962951574395749898329460, 6.47120282126823491157982802144, 6.77156821555178630858224275211, 8.219609508969567811431764434637, 9.374723244681234765771124099752, 10.00344709845432443027923473254
