| L(s) = 1 | + (−1 + i)3-s + (2 − 2i)5-s + (2 + 2i)7-s + i·9-s + (1 + i)11-s + 2·13-s + 4i·15-s + (−4 − i)17-s − 4i·19-s − 4·21-s + (−4 − 4i)23-s − 3i·25-s + (−4 − 4i)27-s + (6 − 6i)29-s + (−6 + 6i)31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.894 − 0.894i)5-s + (0.755 + 0.755i)7-s + 0.333i·9-s + (0.301 + 0.301i)11-s + 0.554·13-s + 1.03i·15-s + (−0.970 − 0.242i)17-s − 0.917i·19-s − 0.872·21-s + (−0.834 − 0.834i)23-s − 0.600i·25-s + (−0.769 − 0.769i)27-s + (1.11 − 1.11i)29-s + (−1.07 + 1.07i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07405 + 0.227765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07405 + 0.227765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (4 + i)T \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2 + 2i)T - 5iT^{2} \) |
| 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (4 + 4i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6 + 6i)T - 29iT^{2} \) |
| 31 | \( 1 + (6 - 6i)T - 31iT^{2} \) |
| 37 | \( 1 + (8 - 8i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 + (4 + 4i)T + 61iT^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 + (-4 - 4i)T + 79iT^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−13.31143634827532823143007410294, −12.17041539856394843093596484185, −11.24876756436953267215165874930, −10.23089008785348678323412894650, −9.100900418652268888806760474411, −8.341878165967709732250305975147, −6.44272941279775910592960789692, −5.24307603797811732552352746845, −4.60285408387943594428588493632, −2.02218370411221792134542555067,
1.69689616983828418430720355533, 3.80644469410721375548118373424, 5.70913527715080602288711552732, 6.52891240013108715853552633225, 7.52782824756656516030151549905, 9.010762990122238157370822239586, 10.38283417544277296419306950370, 11.03109013000032081372752797166, 12.04853780772347251702904442972, 13.28533952118508372847212185166
