| L(s) = 1 | + (−15.4 − 2.11i)3-s + 68.1i·5-s − 49i·7-s + (234. + 65.2i)9-s + 164.·11-s − 181.·13-s + (144. − 1.05e3i)15-s − 68.1i·17-s − 1.79e3i·19-s + (−103. + 756. i)21-s − 435.·23-s − 1.52e3·25-s + (−3.47e3 − 1.50e3i)27-s + 1.91e3i·29-s − 3.13e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.990 − 0.135i)3-s + 1.21i·5-s − 0.377i·7-s + (0.963 + 0.268i)9-s + 0.408·11-s − 0.297·13-s + (0.165 − 1.20i)15-s − 0.0571i·17-s − 1.13i·19-s + (−0.0512 + 0.374i)21-s − 0.171·23-s − 0.486·25-s + (−0.917 − 0.396i)27-s + 0.422i·29-s − 0.585i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.317880100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317880100\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 + 2.11i)T \) |
| 7 | \( 1 + 49iT \) |
| good | 5 | \( 1 - 68.1iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 164.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 181.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 68.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 435.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.91e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.13e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 8.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.36e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.46e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.11e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.13e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.30e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 9.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.92e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.76e5T + 8.58e9T^{2} \) |
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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
−10.78851588572999180351736321974, −10.22575950570640284392317857529, −9.076143172295841621584534044280, −7.46009462212671103466450691719, −6.94531728729951037439225289903, −6.07791720014669724145104536802, −4.88518924322396720284935677494, −3.68193453313171635423274833841, −2.27360740023662705718924595791, −0.68287508467710359127629515839,
0.66073879937205184092770766098, 1.76145115906031536409212233675, 3.80990341121861425805622519268, 4.86046315160233741682798751161, 5.60186515336846247414000396798, 6.59633036617142515164618935837, 7.88005445769444917964036257140, 8.905789757291997463683701148058, 9.753423773055583659352132381349, 10.67589153243394362395800935917
