| L(s) = 1 | + (−4.89 + 2.82i)2-s + (−19.8 + 11.4i)3-s + (15.9 − 27.7i)4-s + (99.9 + 173. i)5-s + (64.9 − 112. i)6-s + 131.·7-s + 181. i·8-s + (−101. + 175. i)9-s + (−979. − 565. i)10-s − 740.·11-s + 734. i·12-s + (−1.61e3 − 933. i)13-s + (−643. + 371. i)14-s + (−3.97e3 − 2.29e3i)15-s + (−512. − 886. i)16-s + (−146. − 254. i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.736 + 0.425i)3-s + (0.249 − 0.433i)4-s + (0.799 + 1.38i)5-s + (0.300 − 0.520i)6-s + 0.383·7-s + 0.353i·8-s + (−0.138 + 0.240i)9-s + (−0.979 − 0.565i)10-s − 0.556·11-s + 0.425i·12-s + (−0.735 − 0.424i)13-s + (−0.234 + 0.135i)14-s + (−1.17 − 0.679i)15-s + (−0.125 − 0.216i)16-s + (−0.0299 − 0.0517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0622321 - 0.572333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0622321 - 0.572333i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 19 | \( 1 + (5.09e3 + 4.58e3i)T \) |
| good | 3 | \( 1 + (19.8 - 11.4i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-99.9 - 173. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 - 131.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 740.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (1.61e3 + 933. i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (146. + 254. i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (8.74e3 - 1.51e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.75e4 - 1.01e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 3.36e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.96e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (2.53e4 - 1.46e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.30e4 - 2.25e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-3.99e4 + 6.91e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.20e5 + 6.96e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.04e5 - 6.06e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.81e5 - 3.13e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.86e5 - 2.23e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-3.45e5 + 1.99e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-2.60e5 - 4.51e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (5.89e5 - 3.40e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 7.06e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-8.96e5 - 5.17e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-1.34e6 + 7.78e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−15.67572471940782307632310822921, −14.73232709206867318453338233250, −13.57029684744119821940490228080, −11.48814157034614495830941071287, −10.57890034396809662489280697597, −9.820572607400638389116535566237, −7.84008653369774078679722669950, −6.43906030480348802623461372516, −5.22542496224375270090719464365, −2.43438143973389611894490039219,
0.36136860725363034160675944416, 1.85511204593162294668419589405, 4.85208574973578527131886161724, 6.29051741487211148594464818153, 8.176198803028115627043091551243, 9.323092978516575173676339595959, 10.64175596584852112334384072506, 12.23298286782172732107292129687, 12.61325357821301538773195046749, 14.20600276077444189191998319300
