| L(s) = 1 | − 2-s + 3.43·3-s + 4-s + 5-s − 3.43·6-s + 7-s − 8-s + 8.79·9-s − 10-s + 3.43·12-s + 5.79·13-s − 14-s + 3.43·15-s + 16-s − 2.43·17-s − 8.79·18-s − 8.22·19-s + 20-s + 3.43·21-s + 0.566·23-s − 3.43·24-s + 25-s − 5.79·26-s + 19.8·27-s + 28-s − 2·29-s − 3.43·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.98·3-s + 0.5·4-s + 0.447·5-s − 1.40·6-s + 0.377·7-s − 0.353·8-s + 2.93·9-s − 0.316·10-s + 0.991·12-s + 1.60·13-s − 0.267·14-s + 0.886·15-s + 0.250·16-s − 0.590·17-s − 2.07·18-s − 1.88·19-s + 0.223·20-s + 0.749·21-s + 0.118·23-s − 0.700·24-s + 0.200·25-s − 1.13·26-s + 3.82·27-s + 0.188·28-s − 0.371·29-s − 0.626·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.219898336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.219898336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 23 | \( 1 - 0.566T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 - 8.86T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 0.433T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 + 5.30T + 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 + 0.0772T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 0.433T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 - 9.22T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
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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
−8.137156215533962079703550526444, −7.37121581688189992928727647436, −6.60652599938840931001023071865, −6.06048093847197179914910908461, −4.74015876446712083699065117933, −3.92760077041846621317712719742, −3.44064178517377647532373699189, −2.26244031483758104768705744688, −2.04946629235157553182033504513, −1.07845614352449133720384295587,
1.07845614352449133720384295587, 2.04946629235157553182033504513, 2.26244031483758104768705744688, 3.44064178517377647532373699189, 3.92760077041846621317712719742, 4.74015876446712083699065117933, 6.06048093847197179914910908461, 6.60652599938840931001023071865, 7.37121581688189992928727647436, 8.137156215533962079703550526444
