L(s) = 1 | + (5.29 − 9.16i)5-s + (−27.6 − 47.8i)11-s + 83.5·13-s + (47.6 + 82.4i)17-s + (−41.7 + 72.3i)19-s + (−82.8 + 143. i)23-s + (6.5 + 11.2i)25-s − 110.·29-s + (41.7 + 72.3i)31-s + (−39 + 67.5i)37-s − 412.·41-s + 148·43-s + (−232. + 403. i)47-s + (55.2 + 95.6i)53-s − 584.·55-s + ⋯ |
L(s) = 1 | + (0.473 − 0.819i)5-s + (−0.757 − 1.31i)11-s + 1.78·13-s + (0.679 + 1.17i)17-s + (−0.504 + 0.873i)19-s + (−0.751 + 1.30i)23-s + (0.0520 + 0.0900i)25-s − 0.707·29-s + (0.241 + 0.419i)31-s + (−0.173 + 0.300i)37-s − 1.57·41-s + 0.524·43-s + (−0.722 + 1.25i)47-s + (0.143 + 0.247i)53-s − 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.933522199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.933522199\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-5.29 + 9.16i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (27.6 + 47.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-47.6 - 82.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (41.7 - 72.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (82.8 - 143. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-41.7 - 72.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (39 - 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 148T + 7.95e4T^{2} \) |
| 47 | \( 1 + (232. - 403. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-55.2 - 95.6i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-275. - 476. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-292. + 506. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-130 - 225. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (334. + 578. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (332 - 575. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 126.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-439. + 760. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.827448076884991061542447114966, −8.355452166273843287885406505462, −7.80457980341520387542087255544, −6.30124907194867327762368835899, −5.83278697777475912751048087695, −5.23940260668858787936248658016, −3.82998691976068670466600261497, −3.36351095515351381069009291995, −1.72473932753797834250155033640, −1.08523970694121390581112336426,
0.43124841097577657604020987384, 1.93161151170237376779680737104, 2.66605454089611740625208235653, 3.72517564349296187282290473785, 4.75106872049467862560343791775, 5.62366125211888531044883238618, 6.60074155280246165665086450747, 7.02192712852317932062542921594, 8.064447004648204766415152744370, 8.773981519149772267367690149940