| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 6·13-s + 14-s − 16-s + 6·17-s + 6·23-s + 6·26-s + 28-s − 6·29-s − 5·32-s − 6·34-s − 4·37-s − 6·41-s − 8·43-s − 6·46-s − 10·47-s + 49-s + 6·52-s + 4·53-s − 3·56-s + 6·58-s − 12·59-s + 10·61-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s + 1.25·23-s + 1.17·26-s + 0.188·28-s − 1.11·29-s − 0.883·32-s − 1.02·34-s − 0.657·37-s − 0.937·41-s − 1.21·43-s − 0.884·46-s − 1.45·47-s + 1/7·49-s + 0.832·52-s + 0.549·53-s − 0.400·56-s + 0.787·58-s − 1.56·59-s + 1.28·61-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.48578808476197, −13.09478800788485, −12.65969079590172, −12.17665308689173, −11.73079151859340, −11.18887195690905, −10.46702341200689, −10.20218212851641, −9.734422302815738, −9.394245949131957, −8.980348277849901, −8.284959601931001, −7.934313556346198, −7.411549890224783, −6.990205031825637, −6.570093083322766, −5.596723699899412, −5.173844442701077, −4.976815438140403, −4.201026024432490, −3.546834735791575, −3.102751293977418, −2.430763118449834, −1.576356772234285, −1.177458367059596, 0, 0,
1.177458367059596, 1.576356772234285, 2.430763118449834, 3.102751293977418, 3.546834735791575, 4.201026024432490, 4.976815438140403, 5.173844442701077, 5.596723699899412, 6.570093083322766, 6.990205031825637, 7.411549890224783, 7.934313556346198, 8.284959601931001, 8.980348277849901, 9.394245949131957, 9.734422302815738, 10.20218212851641, 10.46702341200689, 11.18887195690905, 11.73079151859340, 12.17665308689173, 12.65969079590172, 13.09478800788485, 13.48578808476197
