L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 7-s + 9-s − 4·12-s − 2·13-s + 2·14-s − 4·16-s + 5·17-s − 2·18-s + 8·19-s + 2·21-s + 23-s + 4·26-s + 4·27-s − 2·28-s − 5·29-s − 5·31-s + 8·32-s − 10·34-s + 2·36-s − 7·37-s − 16·38-s + 4·39-s − 7·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 0.377·7-s + 1/3·9-s − 1.15·12-s − 0.554·13-s + 0.534·14-s − 16-s + 1.21·17-s − 0.471·18-s + 1.83·19-s + 0.436·21-s + 0.208·23-s + 0.784·26-s + 0.769·27-s − 0.377·28-s − 0.928·29-s − 0.898·31-s + 1.41·32-s − 1.71·34-s + 1/3·36-s − 1.15·37-s − 2.59·38-s + 0.640·39-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.08956543988940995050722800462, −9.654091189222631032501827269352, −8.654319095271452301047967713911, −7.53022694834992702662478343447, −7.00155655462405775455941547341, −5.73625739555288291436469619815, −5.01098694924706036592409992924, −3.25732449518046581438468298305, −1.39515859044792359909937567271, 0,
1.39515859044792359909937567271, 3.25732449518046581438468298305, 5.01098694924706036592409992924, 5.73625739555288291436469619815, 7.00155655462405775455941547341, 7.53022694834992702662478343447, 8.654319095271452301047967713911, 9.654091189222631032501827269352, 10.08956543988940995050722800462