| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s − 15-s + 16-s − 3·17-s + 18-s + 19-s − 20-s + 22-s − 4·23-s + 24-s − 4·25-s + 27-s + 2·29-s − 30-s − 3·31-s + 32-s + 33-s − 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.538·31-s + 0.176·32-s + 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.821361201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.821361201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.26005328823459, −13.79015527695108, −13.26973223166463, −12.99697640034573, −12.20911212936942, −11.79259507212839, −11.48703151564928, −10.79470245392104, −10.17919478069823, −9.773482270795930, −9.088633026530838, −8.508689482913189, −8.076814404883129, −7.463666603248127, −6.979131747828727, −6.405801459000587, −5.834572877899439, −5.152255549443686, −4.524745602463012, −3.966058821319980, −3.575114222144188, −2.869286453495708, −2.145113058452889, −1.637111363531948, −0.5444353174692817,
0.5444353174692817, 1.637111363531948, 2.145113058452889, 2.869286453495708, 3.575114222144188, 3.966058821319980, 4.524745602463012, 5.152255549443686, 5.834572877899439, 6.405801459000587, 6.979131747828727, 7.463666603248127, 8.076814404883129, 8.508689482913189, 9.088633026530838, 9.773482270795930, 10.17919478069823, 10.79470245392104, 11.48703151564928, 11.79259507212839, 12.20911212936942, 12.99697640034573, 13.26973223166463, 13.79015527695108, 14.26005328823459
