L(s) = 1 | + i·2-s + (−0.597 + 1.62i)3-s − 4-s + 5-s + (−1.62 − 0.597i)6-s − 3.80i·7-s − i·8-s + (−2.28 − 1.94i)9-s + i·10-s + 3.48·11-s + (0.597 − 1.62i)12-s + 3.19·13-s + 3.80·14-s + (−0.597 + 1.62i)15-s + 16-s + 4.76·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.344 + 0.938i)3-s − 0.5·4-s + 0.447·5-s + (−0.663 − 0.243i)6-s − 1.43i·7-s − 0.353i·8-s + (−0.762 − 0.647i)9-s + 0.316i·10-s + 1.05·11-s + (0.172 − 0.469i)12-s + 0.887·13-s + 1.01·14-s + (−0.154 + 0.419i)15-s + 0.250·16-s + 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27673 + 0.743234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27673 + 0.743234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.597 - 1.62i)T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (0.785 - 4.73i)T \) |
good | 7 | \( 1 + 3.80iT - 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 1.88iT - 19T^{2} \) |
| 29 | \( 1 + 7.50iT - 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 4.88iT - 37T^{2} \) |
| 41 | \( 1 - 3.99iT - 41T^{2} \) |
| 43 | \( 1 + 6.96iT - 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 7.35iT - 59T^{2} \) |
| 61 | \( 1 - 3.31iT - 61T^{2} \) |
| 67 | \( 1 + 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 9.92iT - 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 2.98iT - 97T^{2} \) |
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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
−10.27639579467512536826251539734, −9.892939138186215220695343672612, −8.984993916156491066891093034278, −7.989784867378062328006446554028, −6.96136340056219602581744932160, −6.09235729548157655915273313898, −5.31158641419261477547347900056, −3.99287019926624776364639031332, −3.66691091851087366796110374673, −1.07696770045772550266224497628,
1.24011701488949410314518026161, 2.27580669699279282432018471717, 3.38923427573197413764750730317, 5.03258156674706163770201247041, 5.88397082582518152258211743064, 6.53631691020063481465734647499, 7.84981553383223947348695449105, 8.888284166615562724552572464292, 9.206204986578723530113504745037, 10.57958196928520766951277835460