| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s − 2·7-s + 3·8-s + 9-s − 10-s + 2·12-s − 13-s + 2·14-s − 2·15-s − 16-s − 5·17-s − 18-s − 20-s + 4·21-s + 2·23-s − 6·24-s − 4·25-s + 26-s + 4·27-s + 2·28-s − 9·29-s + 2·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.223·20-s + 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s + 0.196·26-s + 0.769·27-s + 0.377·28-s − 1.67·29-s + 0.365·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−14.90219740091354, −14.49246666418927, −13.63188422543312, −13.39565312394578, −12.85030652227572, −12.42149193821497, −11.67903721001109, −11.12568595522060, −10.80170787759229, −10.21701223009512, −9.585692929478664, −9.317995128488986, −8.815501512923683, −8.038052526685100, −7.456489229566413, −6.887714321045415, −6.152615734280098, −5.930822355820645, −5.127676409573777, −4.657258538112954, −4.034314484411378, −3.220772188050061, −2.295765169482099, −1.573719664015643, −0.5915192425615563, 0,
0.5915192425615563, 1.573719664015643, 2.295765169482099, 3.220772188050061, 4.034314484411378, 4.657258538112954, 5.127676409573777, 5.930822355820645, 6.152615734280098, 6.887714321045415, 7.456489229566413, 8.038052526685100, 8.815501512923683, 9.317995128488986, 9.585692929478664, 10.21701223009512, 10.80170787759229, 11.12568595522060, 11.67903721001109, 12.42149193821497, 12.85030652227572, 13.39565312394578, 13.63188422543312, 14.49246666418927, 14.90219740091354
