| L(s) = 1 | − 4.79·2-s + 3·3-s + 14.9·4-s + 12.3·5-s − 14.3·6-s + 26.5·7-s − 33.4·8-s + 9·9-s − 58.9·10-s + 11·11-s + 44.9·12-s − 19.5·13-s − 127.·14-s + 36.9·15-s + 40.4·16-s + 48.6·17-s − 43.1·18-s + 152.·19-s + 184.·20-s + 79.5·21-s − 52.7·22-s + 145.·23-s − 100.·24-s + 26.4·25-s + 93.8·26-s + 27·27-s + 396.·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.87·4-s + 1.10·5-s − 0.978·6-s + 1.43·7-s − 1.47·8-s + 0.333·9-s − 1.86·10-s + 0.301·11-s + 1.08·12-s − 0.417·13-s − 2.42·14-s + 0.635·15-s + 0.632·16-s + 0.694·17-s − 0.564·18-s + 1.84·19-s + 2.06·20-s + 0.826·21-s − 0.510·22-s + 1.31·23-s − 0.853·24-s + 0.211·25-s + 0.707·26-s + 0.192·27-s + 2.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.209084951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.209084951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 4.79T + 8T^{2} \) |
| 5 | \( 1 - 12.3T + 125T^{2} \) |
| 7 | \( 1 - 26.5T + 343T^{2} \) |
| 13 | \( 1 + 19.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 255.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 15.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 1.66T + 3.00e5T^{2} \) |
| 71 | \( 1 - 690.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 849.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 606.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 570.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 455.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 136.T + 9.12e5T^{2} \) |
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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
−8.955018581238506311277237777742, −8.099160575739158775371278433107, −7.47910669928077790616386804868, −6.99097659310972484331510033925, −5.61761582499753401579587897037, −5.06146402278221790544594161447, −3.48237714939422275224289782984, −2.27535938283669315859337298324, −1.64024496866228544856128257427, −0.941906301457121273491937966537,
0.941906301457121273491937966537, 1.64024496866228544856128257427, 2.27535938283669315859337298324, 3.48237714939422275224289782984, 5.06146402278221790544594161447, 5.61761582499753401579587897037, 6.99097659310972484331510033925, 7.47910669928077790616386804868, 8.099160575739158775371278433107, 8.955018581238506311277237777742
