| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 2·13-s − 14-s + 15-s + 16-s − 5·17-s − 18-s − 4·19-s − 20-s − 21-s + 2·22-s − 6·23-s + 24-s + 25-s − 2·26-s − 27-s + 28-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 − 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1517825534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1517825534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.71060192359715, −12.09158351262355, −11.72268262149874, −11.28690498108793, −10.81072722616489, −10.55876403940023, −10.07654149147496, −9.557086128378097, −8.928333772974708, −8.394986241113687, −8.258324483235232, −7.701451426540728, −7.089979011773586, −6.584045940722031, −6.330394542744112, −5.671937368764967, −5.147057054080360, −4.540133173656688, −4.134174145934278, −3.578452655820049, −2.775778143336068, −2.224872855210474, −1.706817208204214, −1.007302688075551, −0.1355383775438719,
0.1355383775438719, 1.007302688075551, 1.706817208204214, 2.224872855210474, 2.775778143336068, 3.578452655820049, 4.134174145934278, 4.540133173656688, 5.147057054080360, 5.671937368764967, 6.330394542744112, 6.584045940722031, 7.089979011773586, 7.701451426540728, 8.258324483235232, 8.394986241113687, 8.928333772974708, 9.557086128378097, 10.07654149147496, 10.55876403940023, 10.81072722616489, 11.28690498108793, 11.72268262149874, 12.09158351262355, 12.71060192359715
