| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s − 2·9-s − 3·10-s + 12-s + 4·13-s − 14-s − 3·15-s + 16-s + 17-s − 2·18-s − 19-s − 3·20-s − 21-s − 6·23-s + 24-s + 4·25-s + 4·26-s − 5·27-s − 28-s − 9·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s − 0.229·19-s − 0.670·20-s − 0.218·21-s − 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.962·27-s − 0.188·28-s − 1.67·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.01263832588515, −13.61960256209100, −13.08860997964299, −12.75058011496073, −12.02191175138386, −11.73201056840886, −11.30014171272837, −10.87057249706233, −10.29453571225755, −9.661036032191494, −8.960182124279318, −8.610658405156546, −8.116554084535119, −7.594691458992965, −7.278285963415352, −6.463594093667272, −6.077476074594496, −5.439177824305049, −4.958185691840503, −4.026154350277025, −3.720241923131196, −3.458919337819912, −2.897080553980471, −1.935898803915725, −1.535004893324657, 0, 0,
1.535004893324657, 1.935898803915725, 2.897080553980471, 3.458919337819912, 3.720241923131196, 4.026154350277025, 4.958185691840503, 5.439177824305049, 6.077476074594496, 6.463594093667272, 7.278285963415352, 7.594691458992965, 8.116554084535119, 8.610658405156546, 8.960182124279318, 9.661036032191494, 10.29453571225755, 10.87057249706233, 11.30014171272837, 11.73201056840886, 12.02191175138386, 12.75058011496073, 13.08860997964299, 13.61960256209100, 14.01263832588515
