| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s − 2·9-s − 10-s − 12-s − 3·13-s − 3·14-s − 15-s + 16-s + 17-s + 2·18-s + 5·19-s + 20-s − 3·21-s − 2·23-s + 24-s + 25-s + 3·26-s + 5·27-s + 3·28-s − 29-s + 30-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 − 2/3·9-s − 0.316·10-s − 0.288·12-s − 0.832·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 1.14·19-s + 0.223·20-s − 0.654·21-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.962·27-s + 0.566·28-s − 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.92252476025395, −14.59279276441875, −13.94288448829675, −13.79760854163921, −12.75266982480515, −12.12563697368967, −11.92279294890110, −11.29449500788580, −10.85834156522560, −10.36251803205229, −9.769278849477315, −9.217177871357145, −8.715884623945681, −8.042994406697624, −7.591449038193841, −7.124131766202529, −6.296003843417544, −5.743151224722050, −5.297115806996156, −4.755656709976111, −3.983731261446440, −2.891477203461101, −2.519375660706143, −1.584038185956983, −0.9955363043970254, 0,
0.9955363043970254, 1.584038185956983, 2.519375660706143, 2.891477203461101, 3.983731261446440, 4.755656709976111, 5.297115806996156, 5.743151224722050, 6.296003843417544, 7.124131766202529, 7.591449038193841, 8.042994406697624, 8.715884623945681, 9.217177871357145, 9.769278849477315, 10.36251803205229, 10.85834156522560, 11.29449500788580, 11.92279294890110, 12.12563697368967, 12.75266982480515, 13.79760854163921, 13.94288448829675, 14.59279276441875, 14.92252476025395
