| L(s) = 1 | − 0.741·3-s + 2.48·5-s − 2.45·9-s − 5.32·11-s − 4.45·13-s − 1.84·15-s + 5.22·17-s − 1.97·19-s − 6.97·23-s + 1.18·25-s + 4.04·27-s + 3.94·29-s − 4.03·31-s + 3.95·33-s + 8.44·37-s + 3.30·39-s + 41-s − 10.5·43-s − 6.09·45-s + 4.67·47-s − 3.87·51-s + 2.15·53-s − 13.2·55-s + 1.46·57-s − 3.06·59-s + 12.0·61-s − 11.0·65-s + ⋯ |
| L(s) = 1 | − 0.428·3-s + 1.11·5-s − 0.816·9-s − 1.60·11-s − 1.23·13-s − 0.475·15-s + 1.26·17-s − 0.452·19-s − 1.45·23-s + 0.236·25-s + 0.777·27-s + 0.733·29-s − 0.725·31-s + 0.687·33-s + 1.38·37-s + 0.528·39-s + 0.156·41-s − 1.61·43-s − 0.908·45-s + 0.681·47-s − 0.542·51-s + 0.295·53-s − 1.78·55-s + 0.193·57-s − 0.399·59-s + 1.53·61-s − 1.37·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.176964620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176964620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.741T + 3T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 3.06T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.549T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.975439820544986503344503450540, −7.13794223108457624057140081449, −6.18959002624797522726301272261, −5.69513148017649999938178698966, −5.25798865631355709290430277060, −4.57165460336451444491569503676, −3.30112809640475482043652817878, −2.50973563147582885577301642767, −2.00560712248042007023899689203, −0.51422496406560420058542713847,
0.51422496406560420058542713847, 2.00560712248042007023899689203, 2.50973563147582885577301642767, 3.30112809640475482043652817878, 4.57165460336451444491569503676, 5.25798865631355709290430277060, 5.69513148017649999938178698966, 6.18959002624797522726301272261, 7.13794223108457624057140081449, 7.975439820544986503344503450540
