| L(s) = 1 | − 4.79i·2-s + (2 − 4.79i)3-s − 14.9·4-s + (−22.9 − 9.59i)6-s + 33.5i·8-s + (−18.9 − 19.1i)9-s + (−29.9 + 71.9i)12-s + 74·13-s + 40.9·16-s + (−91.9 + 91.1i)18-s − 110. i·23-s + (160. + 67.1i)24-s − 125·25-s − 354. i·26-s + (−129. + 52.7i)27-s + ⋯ |
| L(s) = 1 | − 1.69i·2-s + (0.384 − 0.922i)3-s − 1.87·4-s + (−1.56 − 0.652i)6-s + 1.48i·8-s + (−0.703 − 0.710i)9-s + (−0.721 + 1.73i)12-s + 1.57·13-s + 0.640·16-s + (−1.20 + 1.19i)18-s − 0.999i·23-s + (1.36 + 0.571i)24-s − 25-s − 2.67i·26-s + (−0.926 + 0.376i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.384i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.275940 + 1.37859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.275940 + 1.37859i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2 + 4.79i)T \) |
| 23 | \( 1 + 110. iT \) |
| good | 2 | \( 1 + 4.79iT - 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 74T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 29 | \( 1 + 134. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 344T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 306. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 - 642. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 815. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 220. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−13.44994154409642373551389140765, −12.37440759397331378666586814780, −11.56317134080674220664646831262, −10.48819360561422095006563385658, −9.134582039500908511314170686275, −8.134491939867327801130737487398, −6.23143845428292983270152484031, −3.98297866686422702784859810415, −2.53801662311772924551648335151, −0.996139940795100827595483904932,
3.80305279376031239900559755841, 5.22147457768017450620088117595, 6.38010374506982623737782445471, 7.967895682497640834627472920994, 8.748361864258654657959388936913, 9.900023497948015880628023172706, 11.35819148611223138738775427391, 13.45471690948708818404110321292, 14.00130746800366898984595644535, 15.27421098183021062462505979863
