| L(s) = 1 | + 0.309·3-s − 5-s − 4.23·7-s − 2.90·9-s − 2.06·11-s − 4.51·13-s − 0.309·15-s + 4.22·17-s − 8.26·19-s − 1.30·21-s − 2.74·23-s + 25-s − 1.82·27-s + 0.483·29-s − 1.67·31-s − 0.637·33-s + 4.23·35-s + 1.15·37-s − 1.39·39-s + 5.46·41-s − 8.29·43-s + 2.90·45-s − 10.6·47-s + 10.9·49-s + 1.30·51-s − 12.1·53-s + 2.06·55-s + ⋯ |
| L(s) = 1 | + 0.178·3-s − 0.447·5-s − 1.60·7-s − 0.968·9-s − 0.621·11-s − 1.25·13-s − 0.0798·15-s + 1.02·17-s − 1.89·19-s − 0.285·21-s − 0.572·23-s + 0.200·25-s − 0.351·27-s + 0.0897·29-s − 0.301·31-s − 0.110·33-s + 0.715·35-s + 0.190·37-s − 0.223·39-s + 0.852·41-s − 1.26·43-s + 0.432·45-s − 1.55·47-s + 1.56·49-s + 0.182·51-s − 1.67·53-s + 0.278·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03339771992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03339771992\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.309T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 + 8.26T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 0.483T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 4.76T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.96T + 71T^{2} \) |
| 73 | \( 1 + 8.35T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 + 5.38T + 83T^{2} \) |
| 89 | \( 1 + 0.998T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.84173492016777406758680128637, −7.21367555172488656656142345060, −6.28680418041326933207460374124, −6.00283037148402204727512991824, −4.99807687249131490608922090311, −4.26066230471076551318748321154, −3.17134565026192562999490231321, −2.99002817350806663969675538332, −1.97168782006042862972187070206, −0.081605939082843114471797024124,
0.081605939082843114471797024124, 1.97168782006042862972187070206, 2.99002817350806663969675538332, 3.17134565026192562999490231321, 4.26066230471076551318748321154, 4.99807687249131490608922090311, 6.00283037148402204727512991824, 6.28680418041326933207460374124, 7.21367555172488656656142345060, 7.84173492016777406758680128637
