L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s + 12-s − 4·13-s + 15-s − 16-s + 3·17-s + 18-s − 4·19-s + 20-s + 3·24-s + 25-s − 4·26-s − 27-s − 7·29-s + 30-s − 3·31-s + 5·32-s + 3·34-s − 36-s + 3·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s + 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.612·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 1.29·29-s + 0.182·30-s − 0.538·31-s + 0.883·32-s + 0.514·34-s − 1/6·36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.67520288179032, −12.28957613676564, −11.87202561575712, −11.56126353056554, −10.82708353518978, −10.64429193386914, −9.925761662371831, −9.475875540562248, −9.268659953216367, −8.540650346063480, −8.087661047265813, −7.597183787825215, −7.173729773580764, −6.575460856131603, −6.089387038206109, −5.639449895120868, −5.068783258376899, −4.868592694273566, −4.220837083328206, −3.821265107728819, −3.349272993498825, −2.702475476652379, −2.102505621420864, −1.373079015719493, −0.5006336717182887, 0,
0.5006336717182887, 1.373079015719493, 2.102505621420864, 2.702475476652379, 3.349272993498825, 3.821265107728819, 4.220837083328206, 4.868592694273566, 5.068783258376899, 5.639449895120868, 6.089387038206109, 6.575460856131603, 7.173729773580764, 7.597183787825215, 8.087661047265813, 8.540650346063480, 9.268659953216367, 9.475875540562248, 9.925761662371831, 10.64429193386914, 10.82708353518978, 11.56126353056554, 11.87202561575712, 12.28957613676564, 12.67520288179032