L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s − 6·17-s + 4·19-s − 8·23-s + 2·26-s − 2·28-s + 4·31-s − 32-s + 6·34-s − 10·37-s − 4·38-s − 4·41-s + 2·43-s + 8·46-s − 8·47-s − 3·49-s − 2·52-s + 6·53-s + 2·56-s − 10·59-s − 6·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 0.392·26-s − 0.377·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.624·41-s + 0.304·43-s + 1.17·46-s − 1.16·47-s − 3/7·49-s − 0.277·52-s + 0.824·53-s + 0.267·56-s − 1.30·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.68625460600263, −13.81138486640133, −13.71357579044319, −13.11451796661131, −12.36355150222268, −11.99836862960482, −11.69829231081743, −10.88505430302078, −10.47530509010954, −9.983421945001246, −9.517231186699245, −9.076678259283749, −8.538289559959381, −7.911314609483100, −7.507516359585561, −6.753398217691754, −6.494454255480066, −5.925369806707540, −5.156741495881430, −4.620380489305179, −3.842102177675647, −3.269844250264857, −2.601532947661376, −1.987936228203936, −1.307225012402714, 0, 0,
1.307225012402714, 1.987936228203936, 2.601532947661376, 3.269844250264857, 3.842102177675647, 4.620380489305179, 5.156741495881430, 5.925369806707540, 6.494454255480066, 6.753398217691754, 7.507516359585561, 7.911314609483100, 8.538289559959381, 9.076678259283749, 9.517231186699245, 9.983421945001246, 10.47530509010954, 10.88505430302078, 11.69829231081743, 11.99836862960482, 12.36355150222268, 13.11451796661131, 13.71357579044319, 13.81138486640133, 14.68625460600263