| L(s) = 1 | − 2.35·2-s + 0.186·3-s + 3.55·4-s + 2.89·5-s − 0.439·6-s + 7-s − 3.66·8-s − 2.96·9-s − 6.83·10-s − 11-s + 0.663·12-s + 3.08·13-s − 2.35·14-s + 0.540·15-s + 1.53·16-s + 7.05·17-s + 6.98·18-s − 7.24·19-s + 10.3·20-s + 0.186·21-s + 2.35·22-s − 0.384·23-s − 0.684·24-s + 3.40·25-s − 7.28·26-s − 1.11·27-s + 3.55·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.107·3-s + 1.77·4-s + 1.29·5-s − 0.179·6-s + 0.377·7-s − 1.29·8-s − 0.988·9-s − 2.16·10-s − 0.301·11-s + 0.191·12-s + 0.856·13-s − 0.629·14-s + 0.139·15-s + 0.383·16-s + 1.71·17-s + 1.64·18-s − 1.66·19-s + 2.30·20-s + 0.0406·21-s + 0.502·22-s − 0.0801·23-s − 0.139·24-s + 0.680·25-s − 1.42·26-s − 0.214·27-s + 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072045641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072045641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 0.186T + 3T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 + 0.384T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 - 7.98T + 37T^{2} \) |
| 41 | \( 1 - 6.18T + 41T^{2} \) |
| 47 | \( 1 + 4.72T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 - 2.20T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−8.725545170591633679429904064465, −8.053141914673474328969539083489, −7.55296599195218679512478625109, −6.28371015572272578908811920684, −6.01912544346816843486264881398, −5.13139634951743077987230456184, −3.65072395996915949975381697222, −2.44816408184791551205870616454, −1.88758631562660694609535627918, −0.797137276959174457871673325648,
0.797137276959174457871673325648, 1.88758631562660694609535627918, 2.44816408184791551205870616454, 3.65072395996915949975381697222, 5.13139634951743077987230456184, 6.01912544346816843486264881398, 6.28371015572272578908811920684, 7.55296599195218679512478625109, 8.053141914673474328969539083489, 8.725545170591633679429904064465
