| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 13-s + 15-s + 3·17-s − 4·19-s + 2·21-s + 5·23-s + 25-s − 27-s − 9·31-s + 2·35-s + 2·37-s − 39-s + 4·43-s − 45-s + 13·47-s − 3·49-s − 3·51-s − 53-s + 4·57-s − 6·59-s + 13·61-s − 2·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 0.436·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s − 1.61·31-s + 0.338·35-s + 0.328·37-s − 0.160·39-s + 0.609·43-s − 0.149·45-s + 1.89·47-s − 3/7·49-s − 0.420·51-s − 0.137·53-s + 0.529·57-s − 0.781·59-s + 1.66·61-s − 0.251·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.06845852357050, −12.75898291455812, −12.62329834708202, −11.91982497891777, −11.46532781098762, −10.96720704554322, −10.62272148684109, −10.18403206123017, −9.442538595067026, −9.250593759344037, −8.590832645262512, −8.104774553848789, −7.493219236218259, −6.913725420028556, −6.804942710112782, −5.899502225302985, −5.679910113742906, −5.134811883873390, −4.318822355426101, −4.037535885658882, −3.384282606332764, −2.872656273041907, −2.166767991069865, −1.358066672386683, −0.7035506718809189, 0,
0.7035506718809189, 1.358066672386683, 2.166767991069865, 2.872656273041907, 3.384282606332764, 4.037535885658882, 4.318822355426101, 5.134811883873390, 5.679910113742906, 5.899502225302985, 6.804942710112782, 6.913725420028556, 7.493219236218259, 8.104774553848789, 8.590832645262512, 9.250593759344037, 9.442538595067026, 10.18403206123017, 10.62272148684109, 10.96720704554322, 11.46532781098762, 11.91982497891777, 12.62329834708202, 12.75898291455812, 13.06845852357050
