| L(s) = 1 | + (−3.06 + 2.57i)2-s + (−9.33 − 7.83i)3-s + (2.77 − 15.7i)4-s + (75.2 − 27.3i)5-s + 48.7·6-s + (−88.6 + 32.2i)7-s + (32.0 + 55.4i)8-s + (−16.4 − 93.1i)9-s + (−160. + 277. i)10-s + (202. + 350. i)11-s + (−149. + 125. i)12-s + (−8.30 + 47.1i)13-s + (188. − 326. i)14-s + (−916. − 333. i)15-s + (−240. − 87.5i)16-s + (−222. − 1.26e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.598 − 0.502i)3-s + (0.0868 − 0.492i)4-s + (1.34 − 0.490i)5-s + 0.552·6-s + (−0.683 + 0.248i)7-s + (0.176 + 0.306i)8-s + (−0.0675 − 0.383i)9-s + (−0.506 + 0.877i)10-s + (0.504 + 0.874i)11-s + (−0.299 + 0.251i)12-s + (−0.0136 + 0.0773i)13-s + (0.257 − 0.445i)14-s + (−1.05 − 0.382i)15-s + (−0.234 − 0.0855i)16-s + (−0.186 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.469412 - 0.689557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.469412 - 0.689557i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.06 - 2.57i)T \) |
| 37 | \( 1 + (7.37e3 + 3.87e3i)T \) |
| good | 3 | \( 1 + (9.33 + 7.83i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-75.2 + 27.3i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (88.6 - 32.2i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-202. - 350. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (8.30 - 47.1i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (222. + 1.26e3i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 19 | \( 1 + (2.21e3 + 1.85e3i)T + (4.29e5 + 2.43e6i)T^{2} \) |
| 23 | \( 1 + (-1.82e3 + 3.16e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (986. + 1.70e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 3.79e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-2.30e3 + 1.30e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + 5.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.06e4 - 1.84e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.01e4 - 7.33e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (2.39e4 + 8.70e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-5.78e3 + 3.28e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-5.79e4 + 2.10e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (5.45e3 + 4.57e3i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 - 2.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-7.52e4 + 2.73e4i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-341. - 1.93e3i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (2.88e4 + 1.04e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-1.51e3 + 2.62e3i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.07991266165258905508386328669, −12.38788257879895902515198824828, −10.86896374024391902745566311543, −9.459080004296274597165720536856, −9.039944689691751998659166500240, −6.89150653332287297379935342835, −6.32555178366723579305619315541, −5.00367320635072642841039942168, −2.11570570175611174277405758349, −0.45203548532912942077191215219,
1.79975314678321008484586130407, 3.58583556360902411191388742159, 5.63944125164223953624263646132, 6.58854917446278067867014392352, 8.494446497737366930643323363214, 9.816502808850833382138912965226, 10.44442649481514735681618345559, 11.28833631243172908343049604884, 12.87936651791329710081714828246, 13.70330295471527816843256451397
