L(s) = 1 | + (−14.3 − 6.08i)3-s + 63.6i·5-s + 49i·7-s + (169. + 174. i)9-s − 66.5·11-s + 1.15e3·13-s + (387. − 913. i)15-s − 1.56e3i·17-s − 191. i·19-s + (297. − 703. i)21-s − 2.35e3·23-s − 927.·25-s + (−1.36e3 − 3.53e3i)27-s + 3.55e3i·29-s + 3.81e3i·31-s + ⋯ |
L(s) = 1 | + (−0.920 − 0.390i)3-s + 1.13i·5-s + 0.377i·7-s + (0.695 + 0.718i)9-s − 0.165·11-s + 1.89·13-s + (0.444 − 1.04i)15-s − 1.31i·17-s − 0.121i·19-s + (0.147 − 0.348i)21-s − 0.926·23-s − 0.296·25-s + (−0.360 − 0.932i)27-s + 0.785i·29-s + 0.712i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.485804994\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485804994\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (14.3 + 6.08i)T \) |
| 7 | \( 1 - 49iT \) |
good | 5 | \( 1 - 63.6iT - 3.12e3T^{2} \) |
| 11 | \( 1 + 66.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 191. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.55e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.81e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.56e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.55e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 868. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.63e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.34e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.11e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.40e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.05e4T + 8.58e9T^{2} \) |
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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
−10.95895993809912607735451322113, −10.34788487321360412528598295602, −9.059122278572235571283124162890, −7.84232443825944807807236233148, −6.84188150982077173094530780312, −6.18324665240184206646596264022, −5.20579931443414417940541337511, −3.71726904867354289328209385604, −2.42811699083461510180302704418, −0.968567461955962914995757743523,
0.58137921268403749650624262856, 1.50579663452262602182048431649, 3.82554284460635459493457653269, 4.42308390459981755453400575791, 5.78278393984058098056195961406, 6.23780893247043941836558428345, 7.86153229474160476243782828768, 8.702519506311648952977308298966, 9.707267006815236076303691163516, 10.68417409589532851047391746256