| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 11-s + 12-s − 13-s + 16-s − 6·17-s + 2·18-s + 2·19-s + 22-s − 6·23-s − 24-s − 5·25-s + 26-s − 5·27-s + 9·29-s − 4·31-s − 32-s − 33-s + 6·34-s − 2·36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 0.458·19-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 25-s + 0.196·26-s − 0.962·27-s + 1.67·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s + 1.02·34-s − 1/3·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.515724472288571684089286227040, −8.364564560362548777090135115264, −8.265521675023629977633370602556, −7.09577818778194036472308733013, −6.26509669227661442964747384857, −5.23231327938266363285847800045, −3.97281851054077595519569236837, −2.80907868912292619472481812542, −1.95031804062996461477464635472, 0,
1.95031804062996461477464635472, 2.80907868912292619472481812542, 3.97281851054077595519569236837, 5.23231327938266363285847800045, 6.26509669227661442964747384857, 7.09577818778194036472308733013, 8.265521675023629977633370602556, 8.364564560362548777090135115264, 9.515724472288571684089286227040
