| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 12-s − 4·13-s − 4·14-s + 15-s + 16-s − 4·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 8·23-s + 24-s + 25-s + 4·26-s − 27-s + 4·28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.41475065365391, −12.92146660483820, −12.48498989190678, −11.97351362470649, −11.42026946685005, −11.12685982880623, −10.85493130822617, −10.35694935151641, −9.685172449954757, −9.124079944240969, −8.714875058187120, −8.286617559066595, −7.581994931147369, −7.371432416833506, −6.845446339761466, −6.335327491356056, −5.466126065822782, −5.169469942952418, −4.542164292990674, −4.264399992252693, −3.358457475875798, −2.634432210259919, −1.910767217031315, −1.616726087706249, −0.6720017485891130, 0,
0.6720017485891130, 1.616726087706249, 1.910767217031315, 2.634432210259919, 3.358457475875798, 4.264399992252693, 4.542164292990674, 5.169469942952418, 5.466126065822782, 6.335327491356056, 6.845446339761466, 7.371432416833506, 7.581994931147369, 8.286617559066595, 8.714875058187120, 9.124079944240969, 9.685172449954757, 10.35694935151641, 10.85493130822617, 11.12685982880623, 11.42026946685005, 11.97351362470649, 12.48498989190678, 12.92146660483820, 13.41475065365391
