L(s) = 1 | + (0.707 − 0.707i)2-s + (0.204 − 0.204i)3-s − 1.00i·4-s + (1.42 − 1.71i)5-s − 0.289i·6-s + (−0.707 − 0.707i)8-s + 2.91i·9-s + (−0.204 − 2.22i)10-s + 5.62·11-s + (−0.204 − 0.204i)12-s + (1.42 − 1.42i)13-s + (−0.0593 − 0.645i)15-s − 1.00·16-s + (−3.75 − 3.75i)17-s + (2.06 + 2.06i)18-s − 3.89·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.118 − 0.118i)3-s − 0.500i·4-s + (0.639 − 0.768i)5-s − 0.118i·6-s + (−0.250 − 0.250i)8-s + 0.972i·9-s + (−0.0647 − 0.704i)10-s + 1.69·11-s + (−0.0591 − 0.0591i)12-s + (0.396 − 0.396i)13-s + (−0.0153 − 0.166i)15-s − 0.250·16-s + (−0.910 − 0.910i)17-s + (0.486 + 0.486i)18-s − 0.892·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72010 - 1.33345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72010 - 1.33345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.42 + 1.71i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.204 + 0.204i)T - 3iT^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + (-1.42 + 1.42i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.75 + 3.75i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + (0.794 + 0.794i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.15iT - 29T^{2} \) |
| 31 | \( 1 + 3.84iT - 31T^{2} \) |
| 37 | \( 1 + (3.56 - 3.56i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (-1.85 - 1.85i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.16 - 4.16i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.978 + 0.978i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 - 4.60iT - 61T^{2} \) |
| 67 | \( 1 + (-0.597 + 0.597i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + (3.96 - 3.96i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.24iT - 79T^{2} \) |
| 83 | \( 1 + (5.67 - 5.67i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \) |
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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
−10.93446740205964595399948466367, −9.892219730378728685623733437494, −9.064976074282890554489210181318, −8.340913268671095512653830388707, −6.86883122405339740126692446460, −5.99414217633100317597027463150, −4.88418391788126710691491811333, −4.09746346364711286808528342307, −2.49250451162971682008234480879, −1.34152259920313253303084369709,
1.94144264173562908136179055762, 3.57859803272380564418460216231, 4.18328459578121761949037858066, 5.88612450934873064022126917603, 6.48932433528976452069082294330, 7.04932631687402322891216083358, 8.703546297937544300961127572552, 9.115436672668722301227804197157, 10.26183889645691758996382783395, 11.23429398200773465673276905826