| L(s) = 1 | + (3.78 + 11.6i)2-s + (−17.6 + 12.8i)4-s + (−31.6 + 97.5i)5-s + (601. − 436. i)7-s + (1.05e3 + 763. i)8-s − 1.25e3·10-s + (401. − 4.39e3i)11-s + (−2.23e3 − 6.86e3i)13-s + (7.36e3 + 5.34e3i)14-s + (−5.77e3 + 1.77e4i)16-s + (8.33e3 − 2.56e4i)17-s + (−2.08e4 − 1.51e4i)19-s + (−691. − 2.12e3i)20-s + (5.26e4 − 1.19e4i)22-s + 1.03e5·23-s + ⋯ |
| L(s) = 1 | + (0.334 + 1.02i)2-s + (−0.138 + 0.100i)4-s + (−0.113 + 0.348i)5-s + (0.662 − 0.481i)7-s + (0.725 + 0.527i)8-s − 0.396·10-s + (0.0908 − 0.995i)11-s + (−0.281 − 0.866i)13-s + (0.716 + 0.520i)14-s + (−0.352 + 1.08i)16-s + (0.411 − 1.26i)17-s + (−0.696 − 0.506i)19-s + (−0.0193 − 0.0595i)20-s + (1.05 − 0.239i)22-s + 1.77·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.78524 + 0.698663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.78524 + 0.698663i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-401. + 4.39e3i)T \) |
| good | 2 | \( 1 + (-3.78 - 11.6i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (31.6 - 97.5i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-601. + 436. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (2.23e3 + 6.86e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-8.33e3 + 2.56e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (2.08e4 + 1.51e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 1.03e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + (8.63e4 - 6.27e4i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (1.57e4 + 4.85e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.26e5 + 1.64e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-8.65e4 - 6.28e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 3.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.88e4 + 2.09e4i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-3.12e5 - 9.61e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.44e6 - 1.04e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (6.37e4 - 1.96e5i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 - 1.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-6.66e5 + 2.05e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.30e6 + 9.46e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-3.13e5 - 9.66e5i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-1.44e6 + 4.45e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + 5.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.59e6 + 1.41e7i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−12.93993681406291916086765338799, −11.17137177688722191503604121456, −10.81757644510443529071211644280, −9.022750721192527995505591262237, −7.72070734455617937552610229777, −7.04077124866750970841346207831, −5.68517857856024045499749912147, −4.71082983750689833221928449398, −2.92161215952026381761044237236, −0.899730972817475090954806729826,
1.36180133356674205368441554960, 2.32816858560626948194741996134, 3.98165347928533420154600518392, 4.95550101984576313248944583714, 6.76711548390470257845523358858, 8.086949817364933965398463914420, 9.365567008243341565320507198898, 10.55933914580955781261812532398, 11.47789632068106461816433160682, 12.43177855519796080053444315352
