| L(s) = 1 | + (−15.4 − 1.85i)3-s + 55.8i·5-s − 225. i·7-s + (236. + 57.4i)9-s + 36.9·11-s − 795.·13-s + (103. − 864. i)15-s + 129. i·17-s − 998. i·19-s + (−419. + 3.49e3i)21-s + 3.74e3·23-s + 4.87·25-s + (−3.54e3 − 1.32e3i)27-s + 921. i·29-s − 423. i·31-s + ⋯ |
| L(s) = 1 | + (−0.992 − 0.118i)3-s + 0.999i·5-s − 1.74i·7-s + (0.971 + 0.236i)9-s + 0.0920·11-s − 1.30·13-s + (0.118 − 0.992i)15-s + 0.108i·17-s − 0.634i·19-s + (−0.207 + 1.73i)21-s + 1.47·23-s + 0.00156·25-s + (−0.936 − 0.350i)27-s + 0.203i·29-s − 0.0791i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1540986054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1540986054\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 + 1.85i)T \) |
| good | 5 | \( 1 - 55.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 225. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 36.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 795.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 129. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 998. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 921. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 423. iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.44e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.13e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.97e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.15e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−10.32405419117315830216693494132, −9.516981382233408033569159015974, −7.66028506626269063986279002578, −7.08122278622902007384642934144, −6.53352219905044060769583038490, −5.05115725771582317257103068900, −4.20560261893646061388565406126, −2.85665802632792942294332748518, −1.14786310134413275425847705729, −0.05057070788975975033943595436,
1.34677983615161026632003164609, 2.70518595010562865395842425628, 4.54722782595140538826836826496, 5.23113326177491738110924749833, 5.91962372189909282450752261013, 7.12820389820683161679500668482, 8.400797313425543134475181983722, 9.256958244525030245965233466430, 9.919539596913239936173385976474, 11.25507397396957600770429345581
