L(s) = 1 | − 3-s − 5-s − 4.20·7-s + 9-s − 2.75·11-s − 0.753·13-s + 15-s − 4.96·17-s − 4.75·19-s + 4.20·21-s − 23-s + 25-s − 27-s + 4.96·29-s + 0.209·31-s + 2.75·33-s + 4.20·35-s − 5.71·37-s + 0.753·39-s − 9.38·41-s − 12.4·43-s − 45-s − 7.17·47-s + 10.7·49-s + 4.96·51-s − 9.38·53-s + 2.75·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.59·7-s + 0.333·9-s − 0.830·11-s − 0.208·13-s + 0.258·15-s − 1.20·17-s − 1.09·19-s + 0.918·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.921·29-s + 0.0375·31-s + 0.479·33-s + 0.711·35-s − 0.939·37-s + 0.120·39-s − 1.46·41-s − 1.89·43-s − 0.149·45-s − 1.04·47-s + 1.53·49-s + 0.694·51-s − 1.28·53-s + 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1298868796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1298868796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 + 0.753T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 - 0.209T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 - 4.96T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 - 0.209T + 67T^{2} \) |
| 71 | \( 1 + 9.38T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.249168274403541843662902930521, −7.23447779617899550283364496424, −6.44200395169925744241350885666, −6.42757920877965860724614371320, −5.16592220236755315666087666599, −4.60486045149467756356698393001, −3.61996602786703754091005141598, −2.94862976774238059131393036388, −1.91550436238486790551727118414, −0.18496868059260596846442361952,
0.18496868059260596846442361952, 1.91550436238486790551727118414, 2.94862976774238059131393036388, 3.61996602786703754091005141598, 4.60486045149467756356698393001, 5.16592220236755315666087666599, 6.42757920877965860724614371320, 6.44200395169925744241350885666, 7.23447779617899550283364496424, 8.249168274403541843662902930521