| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s − 2·9-s − 11-s + 12-s − 14-s + 16-s + 3·17-s + 2·18-s − 4·19-s + 21-s + 22-s − 6·23-s − 24-s − 5·25-s − 5·27-s + 28-s − 6·29-s − 4·31-s − 32-s − 33-s − 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.917·19-s + 0.218·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 25-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s − 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8512658432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8512658432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.27154467461369, −14.79067259029929, −14.37597861191152, −13.83241653027629, −13.17364183371871, −12.69247226481443, −11.77904568454852, −11.62647019366536, −10.96724380132432, −10.21398110192690, −9.927507522841237, −9.204398892302226, −8.632616266858635, −8.209994452672773, −7.703824025219798, −7.242586042582796, −6.296213136331392, −5.790486642981835, −5.258678807693646, −4.248279014056491, −3.605613888755057, −2.974193267822749, −1.983535736900882, −1.815284523482072, −0.3667216910978856,
0.3667216910978856, 1.815284523482072, 1.983535736900882, 2.974193267822749, 3.605613888755057, 4.248279014056491, 5.258678807693646, 5.790486642981835, 6.296213136331392, 7.242586042582796, 7.703824025219798, 8.209994452672773, 8.632616266858635, 9.204398892302226, 9.927507522841237, 10.21398110192690, 10.96724380132432, 11.62647019366536, 11.77904568454852, 12.69247226481443, 13.17364183371871, 13.83241653027629, 14.37597861191152, 14.79067259029929, 15.27154467461369
