| L(s) = 1 | + (2 − 2i)3-s + (1 − 2i)5-s − 5i·9-s + (−2 + 2i)11-s + 2i·13-s + (−2 − 6i)15-s + (−1 + 4i)17-s + (4 + 4i)23-s + (−3 − 4i)25-s + (−4 − 4i)27-s + (−5 − 5i)29-s + (4 + 4i)31-s + 8i·33-s + (5 − 5i)37-s + (4 + 4i)39-s + ⋯ |
| L(s) = 1 | + (1.15 − 1.15i)3-s + (0.447 − 0.894i)5-s − 1.66i·9-s + (−0.603 + 0.603i)11-s + 0.554i·13-s + (−0.516 − 1.54i)15-s + (−0.242 + 0.970i)17-s + (0.834 + 0.834i)23-s + (−0.600 − 0.800i)25-s + (−0.769 − 0.769i)27-s + (−0.928 − 0.928i)29-s + (0.718 + 0.718i)31-s + 1.39i·33-s + (0.821 − 0.821i)37-s + (0.640 + 0.640i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50414 - 1.23076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50414 - 1.23076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 17 | \( 1 + (1 - 4i)T \) |
| good | 3 | \( 1 + (-2 + 2i)T - 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + (2 - 2i)T - 11iT^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-4 - 4i)T + 23iT^{2} \) |
| 29 | \( 1 + (5 + 5i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4 - 4i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 + (5 - 5i)T - 41iT^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + (-3 + 3i)T - 61iT^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + (8 + 8i)T + 71iT^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 + (12 - 12i)T - 79iT^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−11.64650466670953843326942694463, −10.14251521282321095822423014589, −9.192853618949269026270932781663, −8.493193022464883714532002167431, −7.66716419967769572899405012591, −6.74293581677791942289407042664, −5.49755205257844637318107156366, −4.07081328635572774399498376856, −2.47693977275187017681598269694, −1.48262641894216461678162252850,
2.63300073476749392889035075182, 3.19258874287381682468809072807, 4.56328706157006294660765678262, 5.72283295436841402428248141042, 7.13267249907464647116829217129, 8.161850327567406831073961451050, 9.092999992511108346274323937314, 9.864990421396092352765843891933, 10.63423974739071915194923018869, 11.28992739397175101097663898851
