| L(s) = 1 | − 2·3-s − 62·7-s − 239·9-s + 144·11-s + 654·13-s + 1.19e3·17-s − 556·19-s + 124·21-s + 2.18e3·23-s + 964·27-s − 1.57e3·29-s − 9.66e3·31-s − 288·33-s + 3.53e3·37-s − 1.30e3·39-s + 7.46e3·41-s − 7.11e3·43-s − 2.82e4·47-s − 1.29e4·49-s − 2.38e3·51-s + 1.30e4·53-s + 1.11e3·57-s + 3.70e4·59-s + 3.95e4·61-s + 1.48e4·63-s − 5.67e4·67-s − 4.36e3·69-s + ⋯ |
| L(s) = 1 | − 0.128·3-s − 0.478·7-s − 0.983·9-s + 0.358·11-s + 1.07·13-s + 0.998·17-s − 0.353·19-s + 0.0613·21-s + 0.860·23-s + 0.254·27-s − 0.348·29-s − 1.80·31-s − 0.0460·33-s + 0.424·37-s − 0.137·39-s + 0.693·41-s − 0.586·43-s − 1.86·47-s − 0.771·49-s − 0.128·51-s + 0.637·53-s + 0.0453·57-s + 1.38·59-s + 1.36·61-s + 0.470·63-s − 1.54·67-s − 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{5} T^{2} \) |
| 7 | \( 1 + 62 T + p^{5} T^{2} \) |
| 11 | \( 1 - 144 T + p^{5} T^{2} \) |
| 13 | \( 1 - 654 T + p^{5} T^{2} \) |
| 17 | \( 1 - 70 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2182 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1578 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9660 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3534 T + p^{5} T^{2} \) |
| 41 | \( 1 - 182 p T + p^{5} T^{2} \) |
| 43 | \( 1 + 7114 T + p^{5} T^{2} \) |
| 47 | \( 1 + 602 p T + p^{5} T^{2} \) |
| 53 | \( 1 - 13046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 37092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39570 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56734 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45588 T + p^{5} T^{2} \) |
| 73 | \( 1 + 11842 T + p^{5} T^{2} \) |
| 79 | \( 1 + 94216 T + p^{5} T^{2} \) |
| 83 | \( 1 + 31482 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94054 T + p^{5} T^{2} \) |
| 97 | \( 1 + 23714 T + p^{5} T^{2} \) |
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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
−9.989692813511141450339993683423, −9.024699052277019984529178182574, −8.303804835777743731629939768338, −7.10310362091301738149827969858, −6.08743090169946890313767743509, −5.35004904335204626449701179318, −3.83611036343426490962562037922, −2.96355625634573807261837162474, −1.37850013225501632435348380916, 0,
1.37850013225501632435348380916, 2.96355625634573807261837162474, 3.83611036343426490962562037922, 5.35004904335204626449701179318, 6.08743090169946890313767743509, 7.10310362091301738149827969858, 8.303804835777743731629939768338, 9.024699052277019984529178182574, 9.989692813511141450339993683423
