| L(s) = 1 | − 2-s + 0.519·3-s + 4-s + 0.0567·5-s − 0.519·6-s + 0.0958·7-s − 8-s − 2.73·9-s − 0.0567·10-s − 0.790·11-s + 0.519·12-s − 0.0958·14-s + 0.0294·15-s + 16-s − 7.01·17-s + 2.73·18-s − 19-s + 0.0567·20-s + 0.0498·21-s + 0.790·22-s + 5.09·23-s − 0.519·24-s − 4.99·25-s − 2.97·27-s + 0.0958·28-s + 3.62·29-s − 0.0294·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.299·3-s + 0.5·4-s + 0.0253·5-s − 0.212·6-s + 0.0362·7-s − 0.353·8-s − 0.910·9-s − 0.0179·10-s − 0.238·11-s + 0.149·12-s − 0.0256·14-s + 0.00760·15-s + 0.250·16-s − 1.70·17-s + 0.643·18-s − 0.229·19-s + 0.0126·20-s + 0.0108·21-s + 0.168·22-s + 1.06·23-s − 0.106·24-s − 0.999·25-s − 0.572·27-s + 0.0181·28-s + 0.673·29-s − 0.00537·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022308153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022308153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.519T + 3T^{2} \) |
| 5 | \( 1 - 0.0567T + 5T^{2} \) |
| 7 | \( 1 - 0.0958T + 7T^{2} \) |
| 11 | \( 1 + 0.790T + 11T^{2} \) |
| 17 | \( 1 + 7.01T + 17T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 + 9.34T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 - 7.60T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 + 1.32T + 79T^{2} \) |
| 83 | \( 1 - 9.06T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 - 4.89T + 97T^{2} \) |
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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
−8.225689654157199281379991579662, −7.51394852594506031137046237692, −6.49345328635543854348009708905, −6.27817688558859427459965502367, −5.14392829659423353156818155387, −4.46571627345874731800262039358, −3.35317937509330472364672991585, −2.61278140389754421161750070646, −1.91218320940937260353201691018, −0.54914313781671101280063581572,
0.54914313781671101280063581572, 1.91218320940937260353201691018, 2.61278140389754421161750070646, 3.35317937509330472364672991585, 4.46571627345874731800262039358, 5.14392829659423353156818155387, 6.27817688558859427459965502367, 6.49345328635543854348009708905, 7.51394852594506031137046237692, 8.225689654157199281379991579662
