| L(s) = 1 | − 6.83i·2-s − 15.5·3-s + 17.2·4-s + 167.·5-s + 106. i·6-s + 106. i·7-s − 555. i·8-s + 243·9-s − 1.14e3i·10-s + (−6.11 − 1.33e3i)11-s − 268.·12-s − 1.47e3i·13-s + 729.·14-s − 2.61e3·15-s − 2.69e3·16-s + 3.75e3i·17-s + ⋯ |
| L(s) = 1 | − 0.854i·2-s − 0.577·3-s + 0.269·4-s + 1.34·5-s + 0.493i·6-s + 0.311i·7-s − 1.08i·8-s + 0.333·9-s − 1.14i·10-s + (−0.00459 − 0.999i)11-s − 0.155·12-s − 0.672i·13-s + 0.265·14-s − 0.775·15-s − 0.657·16-s + 0.764i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00459 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.00459 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.35028 - 1.34409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35028 - 1.34409i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 15.5T \) |
| 11 | \( 1 + (6.11 + 1.33e3i)T \) |
| good | 2 | \( 1 + 6.83iT - 64T^{2} \) |
| 5 | \( 1 - 167.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 106. iT - 1.17e5T^{2} \) |
| 13 | \( 1 + 1.47e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.75e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.55e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.91e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 3.24e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.78e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.72e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.06e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 2.95e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 9.09e3T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.54e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.38e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.88e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.47e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.03e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 9.27e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 6.89e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.83e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.10e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 2.42e5T + 8.32e11T^{2} \) |
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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
−15.17189540032689400057229721963, −13.45888981724818078116969596579, −12.64301841117545571070721276654, −11.15785775517009553240627454155, −10.41702640140492916643025904026, −9.061473038164104296333828549492, −6.67972816249766551505146030627, −5.45584359107008433903747585958, −2.90269473632836224807737029894, −1.21247151656150849529085846274,
1.93348304084515901400216595309, 5.06554293288893309691207413680, 6.28975518577990873810798360187, 7.33827240239036113357437456698, 9.361737862551782900742975623550, 10.59968270364238382286301296540, 12.05677088243393843887564887041, 13.57643720515330659866828193510, 14.60840413230846123906702162662, 15.88311434284292483074415535192
