L(s) = 1 | + (−2.85 − 4.94i)2-s + (−22.7 + 39.3i)3-s + (47.7 − 82.6i)4-s + (−48.1 + 83.4i)5-s + 259.·6-s + (330. − 572. i)7-s − 1.27e3·8-s + (62.6 + 108. i)9-s + 549.·10-s − 110.·11-s + (2.16e3 + 3.75e3i)12-s + (−5.92e3 + 1.02e4i)13-s − 3.77e3·14-s + (−2.18e3 − 3.78e3i)15-s + (−2.47e3 − 4.27e3i)16-s + (1.52e4 + 2.64e4i)17-s + ⋯ |
L(s) = 1 | + (−0.252 − 0.436i)2-s + (−0.485 + 0.840i)3-s + (0.372 − 0.645i)4-s + (−0.172 + 0.298i)5-s + 0.489·6-s + (0.364 − 0.630i)7-s − 0.880·8-s + (0.0286 + 0.0495i)9-s + 0.173·10-s − 0.0251·11-s + (0.361 + 0.626i)12-s + (−0.748 + 1.29i)13-s − 0.367·14-s + (−0.167 − 0.289i)15-s + (−0.150 − 0.261i)16-s + (0.753 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0409 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0409 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.693928 + 0.666038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.693928 + 0.666038i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (2.81e5 + 1.24e5i)T \) |
good | 2 | \( 1 + (2.85 + 4.94i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (22.7 - 39.3i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (48.1 - 83.4i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-330. + 572. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 110.T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.92e3 - 1.02e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.52e4 - 2.64e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.11e4 - 3.66e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 - 6.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.20e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-2.59e5 + 4.49e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 - 7.63e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.87e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (1.77e5 + 3.06e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-5.86e5 - 1.01e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.25e6 - 2.17e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.04e5 + 1.04e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.35e6 + 2.34e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 5.42e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.13e6 - 1.96e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.14e5 + 5.44e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-4.59e5 - 7.95e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.10e6T + 8.07e13T^{2} \) |
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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
−15.03757734707132164550977506709, −14.35875776375960731994981409190, −12.36101925044442625836351705126, −10.93750023737980797897054889375, −10.55911114022809149492911111292, −9.292625601481363134612862648677, −7.23797256652571238662311614781, −5.57719931206928727378124916953, −4.00722499373603300108452576851, −1.69223452300568406962677624078,
0.48788554055540211673626559134, 2.73550052152640643420785888928, 5.29744496439820711024242489302, 6.86151267222972431492322983891, 7.80098880881353227278860785639, 9.141846752738894225774001390518, 11.23229453607592253996816893001, 12.32077633328920680592582434715, 12.88501384747347327860182159714, 14.85747230325575361485884255616