L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 12-s − 2·13-s − 4·14-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s + 24-s + 25-s − 2·26-s + 27-s − 4·28-s + 6·29-s + 30-s − 8·31-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 − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.306341395\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.306341395\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.69524190471727, −14.09110584012878, −13.69304754286180, −13.26698441879388, −12.67848170172788, −12.53829263142101, −11.71820011663418, −11.16099819465517, −10.47084271348970, −9.950654412083061, −9.545246853378187, −8.992310010233821, −8.496278894752842, −7.588442765248437, −7.038400974279028, −6.675625417492124, −6.101241678312556, −5.457460316167117, −4.796607264067746, −4.158511576407779, −3.434743403919719, −2.989456944997064, −2.382499605960354, −1.721576403737266, −0.5330110556768413,
0.5330110556768413, 1.721576403737266, 2.382499605960354, 2.989456944997064, 3.434743403919719, 4.158511576407779, 4.796607264067746, 5.457460316167117, 6.101241678312556, 6.675625417492124, 7.038400974279028, 7.588442765248437, 8.496278894752842, 8.992310010233821, 9.545246853378187, 9.950654412083061, 10.47084271348970, 11.16099819465517, 11.71820011663418, 12.53829263142101, 12.67848170172788, 13.26698441879388, 13.69304754286180, 14.09110584012878, 14.69524190471727