| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 3·7-s + 8-s + 9-s + 10-s + 12-s − 5·13-s − 3·14-s + 15-s + 16-s − 7·17-s + 18-s − 7·19-s + 20-s − 3·21-s + 24-s + 25-s − 5·26-s + 27-s − 3·28-s + 7·29-s + 30-s + 6·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.38·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.60·19-s + 0.223·20-s − 0.654·21-s + 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s − 0.566·28-s + 1.29·29-s + 0.182·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.301358841691827269298510173902, −7.11530247132256972474858179506, −6.60469694443004686612795967977, −6.17065055117509428916144017199, −4.81558703374609103518008249673, −4.49622811046928333256503569121, −3.33640957951997981299135744561, −2.59890749960671635168085278421, −1.95883093399684154034067499901, 0,
1.95883093399684154034067499901, 2.59890749960671635168085278421, 3.33640957951997981299135744561, 4.49622811046928333256503569121, 4.81558703374609103518008249673, 6.17065055117509428916144017199, 6.60469694443004686612795967977, 7.11530247132256972474858179506, 8.301358841691827269298510173902
