| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s + 6·9-s + 4·10-s − 5·11-s − 6·12-s − 2·13-s + 2·14-s + 6·15-s − 4·16-s − 12·18-s − 4·20-s + 3·21-s + 10·22-s + 2·23-s − 25-s + 4·26-s − 9·27-s − 2·28-s + 6·29-s − 12·30-s − 4·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 1.26·10-s − 1.50·11-s − 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.54·15-s − 16-s − 2.82·18-s − 0.894·20-s + 0.654·21-s + 2.13·22-s + 0.417·23-s − 1/5·25-s + 0.784·26-s − 1.73·27-s − 0.377·28-s + 1.11·29-s − 2.19·30-s − 0.718·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.19201741687448195508003517837, −15.60385787320431886505786442681, −12.95838641388284594767331070614, −11.75732472284977634676677942867, −10.77513816254080044575148887495, −9.933098353605351714295524224830, −8.014330807872879223418899196928, −6.87039121695443194852465216240, −5.00317001400665869534627315571, 0,
5.00317001400665869534627315571, 6.87039121695443194852465216240, 8.014330807872879223418899196928, 9.933098353605351714295524224830, 10.77513816254080044575148887495, 11.75732472284977634676677942867, 12.95838641388284594767331070614, 15.60385787320431886505786442681, 16.19201741687448195508003517837
