| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 5·11-s + 12-s − 2·13-s − 14-s − 15-s + 16-s + 18-s − 2·19-s − 20-s − 21-s + 5·22-s + 4·23-s + 24-s − 4·25-s − 2·26-s + 27-s − 28-s − 9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.218·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.46849874749354, −16.12598413539997, −15.10827000078456, −14.94272538261436, −14.54664815252106, −13.83957506982696, −13.15596599545083, −12.85846889931102, −11.99714036464963, −11.74178230983994, −11.00851910364636, −10.36703875030485, −9.496009491018452, −9.145984024555538, −8.509422966735683, −7.535710143643257, −7.208155160714449, −6.554684210416815, −5.837892704506232, −5.045971352240716, −4.274907024820598, −3.577751625625459, −3.342314573591787, −2.110803346137990, −1.535420107004831, 0,
1.535420107004831, 2.110803346137990, 3.342314573591787, 3.577751625625459, 4.274907024820598, 5.045971352240716, 5.837892704506232, 6.554684210416815, 7.208155160714449, 7.535710143643257, 8.509422966735683, 9.145984024555538, 9.496009491018452, 10.36703875030485, 11.00851910364636, 11.74178230983994, 11.99714036464963, 12.85846889931102, 13.15596599545083, 13.83957506982696, 14.54664815252106, 14.94272538261436, 15.10827000078456, 16.12598413539997, 16.46849874749354
