| L(s) = 1 | + (3.06 + 2.57i)2-s + (−17.2 + 14.4i)3-s + (2.77 + 15.7i)4-s + (−6.83 − 2.48i)5-s − 90.0·6-s + (−141. − 51.3i)7-s + (−32.0 + 55.4i)8-s + (45.7 − 259. i)9-s + (−14.5 − 25.1i)10-s + (270. − 468. i)11-s + (−275. − 231. i)12-s + (187. + 1.06e3i)13-s + (−300. − 520. i)14-s + (153. − 55.9i)15-s + (−240. + 87.5i)16-s + (405. − 2.29e3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−1.10 + 0.928i)3-s + (0.0868 + 0.492i)4-s + (−0.122 − 0.0444i)5-s − 1.02·6-s + (−1.08 − 0.396i)7-s + (−0.176 + 0.306i)8-s + (0.188 − 1.06i)9-s + (−0.0459 − 0.0796i)10-s + (0.673 − 1.16i)11-s + (−0.553 − 0.464i)12-s + (0.308 + 1.74i)13-s + (−0.409 − 0.709i)14-s + (0.176 − 0.0642i)15-s + (−0.234 + 0.0855i)16-s + (0.340 − 1.92i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.205770 - 0.176249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.205770 - 0.176249i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (8.04e3 - 2.16e3i)T \) |
| good | 3 | \( 1 + (17.2 - 14.4i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (6.83 + 2.48i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (141. + 51.3i)T + (1.28e4 + 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-270. + 468. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-187. - 1.06e3i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-405. + 2.29e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 19 | \( 1 + (285. - 239. i)T + (4.29e5 - 2.43e6i)T^{2} \) |
| 23 | \( 1 + (1.43e3 + 2.49e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.46e3 - 4.27e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 430.T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-1.83e3 - 1.04e4i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 - 2.22e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.50e4 + 2.60e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.67e4 - 6.10e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-1.54e4 + 5.63e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-4.38e3 - 2.48e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (4.78e4 + 1.73e4i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (2.93e4 - 2.46e4i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + 5.68e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.65e3 - 965. i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-9.97e3 + 5.65e4i)T + (-3.70e9 - 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-5.10e4 + 1.85e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-5.12e3 - 8.87e3i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.56086215446180127727941718212, −11.94243601403935939616271559599, −11.41199376002390302608504881936, −10.02580976832381542435217988623, −8.944847813703601614314355778667, −6.85271777009714585036124228138, −6.06295318285968108097265331546, −4.64779122171512492648270191225, −3.55159339601321199607930538254, −0.11342207054853649573590268541,
1.62228302494745182819440507649, 3.64069209655432699774343373749, 5.65459429323133654929721853104, 6.28852938419219069720419583694, 7.68158619753398966533015022608, 9.688591747769027959303705545093, 10.75767570946236956202601245548, 12.00866237208012038036590941701, 12.67515891801197397263619153798, 13.17878625969691710741479721495
