L(s) = 1 | − 4·2-s + 16·4-s − 141.·7-s − 64·8-s − 113.·11-s − 61.7·13-s + 566.·14-s + 256·16-s − 1.67e3·17-s − 662.·19-s + 453.·22-s − 86.4·23-s + 246.·26-s − 2.26e3·28-s + 3.23e3·29-s − 3.81e3·31-s − 1.02e3·32-s + 6.68e3·34-s − 1.02e4·37-s + 2.64e3·38-s + 1.34e4·41-s + 4.69e3·43-s − 1.81e3·44-s + 345.·46-s + 1.52e4·47-s + 3.25e3·49-s − 987.·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.09·7-s − 0.353·8-s − 0.282·11-s − 0.101·13-s + 0.772·14-s + 0.250·16-s − 1.40·17-s − 0.420·19-s + 0.199·22-s − 0.0340·23-s + 0.0716·26-s − 0.546·28-s + 0.713·29-s − 0.712·31-s − 0.176·32-s + 0.991·34-s − 1.23·37-s + 0.297·38-s + 1.25·41-s + 0.387·43-s − 0.141·44-s + 0.0241·46-s + 1.00·47-s + 0.193·49-s − 0.0506·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7052849715\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7052849715\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 113.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 61.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.67e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 662.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 86.4T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 498.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.35e4T + 8.58e9T^{2} \) |
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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
−10.30430860287856536412095920278, −9.257374934401568535077012748577, −8.734813616639824872696948552561, −7.50685482881525271668780725221, −6.68709668876032035040045315502, −5.86098021955007628672792460913, −4.40876468403547833817143894168, −3.11754184772086575357561348217, −2.06248025305415981788920729837, −0.45735918814185854214405374326,
0.45735918814185854214405374326, 2.06248025305415981788920729837, 3.11754184772086575357561348217, 4.40876468403547833817143894168, 5.86098021955007628672792460913, 6.68709668876032035040045315502, 7.50685482881525271668780725221, 8.734813616639824872696948552561, 9.257374934401568535077012748577, 10.30430860287856536412095920278