| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s − 12-s + 6·13-s + 14-s + 2·15-s + 16-s − 2·17-s + 18-s − 4·19-s − 2·20-s − 21-s + 23-s − 24-s − 25-s + 6·26-s − 27-s + 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.80961941958706, −13.29501499139478, −12.86349323792031, −12.38078444625509, −11.82428256120043, −11.45098121729473, −11.11911563703877, −10.57154083359163, −10.30089697379180, −9.423596324753226, −8.711481037325074, −8.357410639350573, −7.962043369659768, −7.169973429760237, −6.825190380504035, −6.132706396804790, −5.935439074229244, −5.125990448044742, −4.589322152866420, −4.147017857623332, −3.706064412892258, −3.104398272054865, −2.294998561414088, −1.555282722167616, −0.9171823691271393, 0,
0.9171823691271393, 1.555282722167616, 2.294998561414088, 3.104398272054865, 3.706064412892258, 4.147017857623332, 4.589322152866420, 5.125990448044742, 5.935439074229244, 6.132706396804790, 6.825190380504035, 7.169973429760237, 7.962043369659768, 8.357410639350573, 8.711481037325074, 9.423596324753226, 10.30089697379180, 10.57154083359163, 11.11911563703877, 11.45098121729473, 11.82428256120043, 12.38078444625509, 12.86349323792031, 13.29501499139478, 13.80961941958706
