L(s) = 1 | − 3·9-s − 6·13-s − 2·17-s − 4·29-s + 12·37-s + 8·41-s − 7·49-s − 4·53-s − 12·61-s + 6·73-s + 9·81-s − 10·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
L(s) = 1 | − 9-s − 1.66·13-s − 0.485·17-s − 0.742·29-s + 1.97·37-s + 1.24·41-s − 49-s − 0.549·53-s − 1.53·61-s + 0.702·73-s + 81-s − 1.05·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.19919868897692, −12.88028541630475, −12.40158218420034, −11.84675573803970, −11.53803337899891, −10.84288636374542, −10.79414019696035, −9.839140995402325, −9.490924450263980, −9.261406671309156, −8.569015761705971, −7.962576150057559, −7.664824754937602, −7.198198115590592, −6.416973677896994, −6.170364837427911, −5.444313282859599, −5.091051998145492, −4.430058709949119, −4.073965691673906, −3.141266696399849, −2.745687471722069, −2.299166069871877, −1.596208778583379, −0.6221911259078061, 0,
0.6221911259078061, 1.596208778583379, 2.299166069871877, 2.745687471722069, 3.141266696399849, 4.073965691673906, 4.430058709949119, 5.091051998145492, 5.444313282859599, 6.170364837427911, 6.416973677896994, 7.198198115590592, 7.664824754937602, 7.962576150057559, 8.569015761705971, 9.261406671309156, 9.490924450263980, 9.839140995402325, 10.79414019696035, 10.84288636374542, 11.53803337899891, 11.84675573803970, 12.40158218420034, 12.88028541630475, 13.19919868897692