| L(s) = 1 | − 3.82·2-s − 3·3-s + 6.63·4-s + 0.275·5-s + 11.4·6-s + 0.0981·7-s + 5.20·8-s + 9·9-s − 1.05·10-s − 0.749·11-s − 19.9·12-s − 0.375·14-s − 0.826·15-s − 73.0·16-s + 53.7·17-s − 34.4·18-s + 145.·19-s + 1.82·20-s − 0.294·21-s + 2.86·22-s + 29.3·23-s − 15.6·24-s − 124.·25-s − 27·27-s + 0.651·28-s − 267.·29-s + 3.16·30-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.577·3-s + 0.829·4-s + 0.0246·5-s + 0.781·6-s + 0.00529·7-s + 0.230·8-s + 0.333·9-s − 0.0333·10-s − 0.0205·11-s − 0.479·12-s − 0.00716·14-s − 0.0142·15-s − 1.14·16-s + 0.767·17-s − 0.450·18-s + 1.75·19-s + 0.0204·20-s − 0.00305·21-s + 0.0278·22-s + 0.266·23-s − 0.132·24-s − 0.999·25-s − 0.192·27-s + 0.00439·28-s − 1.71·29-s + 0.0192·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6736646036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6736646036\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 5 | \( 1 - 0.275T + 125T^{2} \) |
| 7 | \( 1 - 0.0981T + 343T^{2} \) |
| 11 | \( 1 + 0.749T + 1.33e3T^{2} \) |
| 17 | \( 1 - 53.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 430.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 495.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 310.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 203.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 636.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 700.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 874.T + 9.12e5T^{2} \) |
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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
−10.22214709207844841795584025189, −9.680214602696116754287965489778, −8.896995555856797845425437451628, −7.66108594470037633652550798885, −7.33362651257343420853676679591, −5.95610191581853071431427226344, −5.01645338434532951041943873691, −3.54921232293870494391195718479, −1.80303940809505988019019324902, −0.65879724760744432710805663340,
0.65879724760744432710805663340, 1.80303940809505988019019324902, 3.54921232293870494391195718479, 5.01645338434532951041943873691, 5.95610191581853071431427226344, 7.33362651257343420853676679591, 7.66108594470037633652550798885, 8.896995555856797845425437451628, 9.680214602696116754287965489778, 10.22214709207844841795584025189
