| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s + 9-s + 2·10-s + 11-s − 2·12-s − 3·13-s + 2·14-s + 15-s − 4·16-s − 2·18-s − 6·19-s − 2·20-s + 21-s − 2·22-s + 3·23-s + 25-s + 6·26-s − 27-s − 2·28-s − 2·30-s + 3·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s − 0.832·13-s + 0.534·14-s + 0.258·15-s − 16-s − 0.471·18-s − 1.37·19-s − 0.447·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s − 0.365·30-s + 0.538·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2323205317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2323205317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.43080564609328, −11.97130922949322, −11.52392178801871, −11.18766537258237, −10.60047804028393, −10.33701051458639, −9.799490207319319, −9.480463492862467, −8.923591905925521, −8.524615357967101, −8.061634632982506, −7.509886044832887, −7.194608945982867, −6.657625925658935, −6.235409654195706, −5.792032894993210, −4.895843163635157, −4.412136227167613, −4.330542784381715, −3.250005553856405, −2.827627176139451, −2.084473377179514, −1.547072352961161, −0.8617526174660022, −0.2144153939632357,
0.2144153939632357, 0.8617526174660022, 1.547072352961161, 2.084473377179514, 2.827627176139451, 3.250005553856405, 4.330542784381715, 4.412136227167613, 4.895843163635157, 5.792032894993210, 6.235409654195706, 6.657625925658935, 7.194608945982867, 7.509886044832887, 8.061634632982506, 8.524615357967101, 8.923591905925521, 9.480463492862467, 9.799490207319319, 10.33701051458639, 10.60047804028393, 11.18766537258237, 11.52392178801871, 11.97130922949322, 12.43080564609328
