| L(s) = 1 | + (−3.75 − 1.36i)2-s + (−2.35 + 0.857i)3-s + (12.2 + 10.2i)4-s + (−1.87 + 10.6i)5-s + 10.0·6-s + (20.8 − 118. i)7-s + (−32.0 − 55.4i)8-s + (−181. + 152. i)9-s + (21.5 − 37.3i)10-s + (101. + 175. i)11-s + (−37.7 − 13.7i)12-s + (259. + 217. i)13-s + (−239. + 415. i)14-s + (−4.70 − 26.6i)15-s + (44.4 + 252. i)16-s + (789. − 662. i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.151 + 0.0550i)3-s + (0.383 + 0.321i)4-s + (−0.0335 + 0.190i)5-s + 0.113·6-s + (0.160 − 0.910i)7-s + (−0.176 − 0.306i)8-s + (−0.746 + 0.626i)9-s + (0.0682 − 0.118i)10-s + (0.252 + 0.436i)11-s + (−0.0755 − 0.0275i)12-s + (0.425 + 0.356i)13-s + (−0.327 + 0.566i)14-s + (−0.00539 − 0.0305i)15-s + (0.0434 + 0.246i)16-s + (0.662 − 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.21246 - 0.116596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21246 - 0.116596i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.75 + 1.36i)T \) |
| 37 | \( 1 + (4.85e3 + 6.76e3i)T \) |
| good | 3 | \( 1 + (2.35 - 0.857i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (1.87 - 10.6i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-20.8 + 118. i)T + (-1.57e4 - 5.74e3i)T^{2} \) |
| 11 | \( 1 + (-101. - 175. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-259. - 217. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-789. + 662. i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 19 | \( 1 + (-1.47e3 + 535. i)T + (1.89e6 - 1.59e6i)T^{2} \) |
| 23 | \( 1 + (-1.30e3 + 2.25e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.53e3 - 4.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 9.79e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-3.73e3 - 3.13e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + 4.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.00e3 - 1.38e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (722. + 4.09e3i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-4.98e3 - 2.82e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (3.03e4 + 2.55e4i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-6.01e3 + 3.40e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (2.37e4 - 8.62e3i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 - 2.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (4.80 - 27.2i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-3.95e4 + 3.31e4i)T + (6.84e8 - 3.87e9i)T^{2} \) |
| 89 | \( 1 + (-1.09e4 - 6.19e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-6.37e4 + 1.10e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.66662718949096195232125404759, −12.17700505992709757063194261786, −11.11423322270681742436903270578, −10.33853870489459265141237690392, −9.036124230036215610424369532500, −7.77816399861596062803612558698, −6.68879191403532601914491283445, −4.82537672990457843917192437706, −2.98223766674636747174826591192, −0.992705388691007164112643409300,
0.995089470240500352264033506874, 3.08568549912627033105024100740, 5.42249044495075022542193676922, 6.38142571773642093707119128635, 8.116059839059154031633942077066, 8.874366419602952162487464326893, 10.08834959530343495713534505445, 11.53365853447050050375327337035, 12.15426612066219247835223109474, 13.76396321399875913833297234391
