L(s) = 1 | − 22·3-s + 25·5-s + 218·7-s + 241·9-s + 480·11-s + 622·13-s − 550·15-s + 186·17-s + 1.20e3·19-s − 4.79e3·21-s − 3.18e3·23-s + 625·25-s + 44·27-s − 5.52e3·29-s + 9.35e3·31-s − 1.05e4·33-s + 5.45e3·35-s − 5.61e3·37-s − 1.36e4·39-s − 1.43e4·41-s + 370·43-s + 6.02e3·45-s + 1.61e4·47-s + 3.07e4·49-s − 4.09e3·51-s + 4.37e3·53-s + 1.20e4·55-s + ⋯ |
L(s) = 1 | − 1.41·3-s + 0.447·5-s + 1.68·7-s + 0.991·9-s + 1.19·11-s + 1.02·13-s − 0.631·15-s + 0.156·17-s + 0.765·19-s − 2.37·21-s − 1.25·23-s + 1/5·25-s + 0.0116·27-s − 1.22·29-s + 1.74·31-s − 1.68·33-s + 0.752·35-s − 0.674·37-s − 1.44·39-s − 1.33·41-s + 0.0305·43-s + 0.443·45-s + 1.06·47-s + 1.82·49-s − 0.220·51-s + 0.213·53-s + 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.000045542\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000045542\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
good | 3 | \( 1 + 22 T + p^{5} T^{2} \) |
| 7 | \( 1 - 218 T + p^{5} T^{2} \) |
| 11 | \( 1 - 480 T + p^{5} T^{2} \) |
| 13 | \( 1 - 622 T + p^{5} T^{2} \) |
| 17 | \( 1 - 186 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1204 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3186 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9356 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5618 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14394 T + p^{5} T^{2} \) |
| 43 | \( 1 - 370 T + p^{5} T^{2} \) |
| 47 | \( 1 - 16146 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11748 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13202 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11542 T + p^{5} T^{2} \) |
| 71 | \( 1 + 29532 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33698 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31208 T + p^{5} T^{2} \) |
| 83 | \( 1 - 38466 T + p^{5} T^{2} \) |
| 89 | \( 1 - 119514 T + p^{5} T^{2} \) |
| 97 | \( 1 - 94658 T + p^{5} T^{2} \) |
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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
−10.98668541700642553582210617146, −10.17185805539943443464370039470, −8.910625215499231818669415826326, −7.911277503459618332483245527739, −6.64260677278629920410481616865, −5.80351499906646376029988529386, −5.00607540471512156675270326931, −3.94334560162021756372403947958, −1.73695807122594335718778322397, −0.938276032071123315674735698199,
0.938276032071123315674735698199, 1.73695807122594335718778322397, 3.94334560162021756372403947958, 5.00607540471512156675270326931, 5.80351499906646376029988529386, 6.64260677278629920410481616865, 7.911277503459618332483245527739, 8.910625215499231818669415826326, 10.17185805539943443464370039470, 10.98668541700642553582210617146