L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 5·7-s − 8-s + 9-s + 3·11-s − 2·12-s + 5·14-s + 16-s + 3·17-s − 18-s + 4·19-s + 10·21-s − 3·22-s + 6·23-s + 2·24-s + 4·27-s − 5·28-s + 9·29-s − 5·31-s − 32-s − 6·33-s − 3·34-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 1.33·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s + 2.18·21-s − 0.639·22-s + 1.25·23-s + 0.408·24-s + 0.769·27-s − 0.944·28-s + 1.67·29-s − 0.898·31-s − 0.176·32-s − 1.04·33-s − 0.514·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6887500659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6887500659\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.56910601760243817093888235345, −6.86413435515224871887119448876, −6.58570439122203097567740225037, −5.83205433123706344564867795722, −5.36390728949521231113059130105, −4.24756367568359228398648081667, −3.25245667982124234269274320629, −2.83355822382816546800333182855, −1.23608255037534884152711631005, −0.55332360837024460141857494917,
0.55332360837024460141857494917, 1.23608255037534884152711631005, 2.83355822382816546800333182855, 3.25245667982124234269274320629, 4.24756367568359228398648081667, 5.36390728949521231113059130105, 5.83205433123706344564867795722, 6.58570439122203097567740225037, 6.86413435515224871887119448876, 7.56910601760243817093888235345