| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 6·17-s + 18-s − 6·19-s − 20-s + 21-s − 22-s − 24-s + 25-s + 26-s − 27-s − 28-s + 9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−12.77859953288794, −12.15017497068511, −11.90906145110719, −11.56457123667806, −10.91241535419016, −10.55869703990461, −10.06177528216256, −9.860229834230557, −9.079356728253046, −8.362044093968669, −8.213654528110884, −7.604299460753948, −7.099166316532854, −6.498033205236069, −6.246663275485836, −5.839062081615554, −5.123297076417190, −4.686786020000239, −4.444792941316229, −3.632633749902566, −3.217023241473172, −2.882429711350669, −1.975052484312490, −1.465033572909192, −0.7215906441753590, 0,
0.7215906441753590, 1.465033572909192, 1.975052484312490, 2.882429711350669, 3.217023241473172, 3.632633749902566, 4.444792941316229, 4.686786020000239, 5.123297076417190, 5.839062081615554, 6.246663275485836, 6.498033205236069, 7.099166316532854, 7.604299460753948, 8.213654528110884, 8.362044093968669, 9.079356728253046, 9.860229834230557, 10.06177528216256, 10.55869703990461, 10.91241535419016, 11.56457123667806, 11.90906145110719, 12.15017497068511, 12.77859953288794
