L(s) = 1 | + 2-s − 1.25·3-s + 4-s − 3.07·5-s − 1.25·6-s − 0.487·7-s + 8-s − 1.43·9-s − 3.07·10-s − 4.26·11-s − 1.25·12-s − 2.00·13-s − 0.487·14-s + 3.84·15-s + 16-s + 4.08·17-s − 1.43·18-s + 8.04·19-s − 3.07·20-s + 0.610·21-s − 4.26·22-s − 0.0257·23-s − 1.25·24-s + 4.45·25-s − 2.00·26-s + 5.54·27-s − 0.487·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.721·3-s + 0.5·4-s − 1.37·5-s − 0.510·6-s − 0.184·7-s + 0.353·8-s − 0.478·9-s − 0.972·10-s − 1.28·11-s − 0.360·12-s − 0.554·13-s − 0.130·14-s + 0.992·15-s + 0.250·16-s + 0.990·17-s − 0.338·18-s + 1.84·19-s − 0.687·20-s + 0.133·21-s − 0.909·22-s − 0.00537·23-s − 0.255·24-s + 0.891·25-s − 0.392·26-s + 1.06·27-s − 0.0922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.204847797\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204847797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 0.487T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 19 | \( 1 - 8.04T + 19T^{2} \) |
| 23 | \( 1 + 0.0257T + 23T^{2} \) |
| 29 | \( 1 - 9.96T + 29T^{2} \) |
| 31 | \( 1 - 0.393T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 + 0.996T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 0.816T + 71T^{2} \) |
| 73 | \( 1 - 1.75T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.773092315051670297265913063777, −8.265863378312774101143761201011, −7.82515039612633227289347210797, −7.06688907687554126937390726100, −6.03910202736533514677273986525, −5.14376397935888652914088191833, −4.71382403638570548833980292151, −3.35181894346442828169522582611, −2.84628103937294037357414424457, −0.70832130357303245934045032779,
0.70832130357303245934045032779, 2.84628103937294037357414424457, 3.35181894346442828169522582611, 4.71382403638570548833980292151, 5.14376397935888652914088191833, 6.03910202736533514677273986525, 7.06688907687554126937390726100, 7.82515039612633227289347210797, 8.265863378312774101143761201011, 9.773092315051670297265913063777