| L(s) = 1 | − 0.589·2-s + 0.0839·3-s − 1.65·4-s − 0.856·5-s − 0.0494·6-s + 3.14·7-s + 2.15·8-s − 2.99·9-s + 0.505·10-s − 0.138·12-s − 1.46·13-s − 1.85·14-s − 0.0719·15-s + 2.03·16-s − 5.02·17-s + 1.76·18-s + 0.707·19-s + 1.41·20-s + 0.263·21-s + 7.30·23-s + 0.180·24-s − 4.26·25-s + 0.862·26-s − 0.502·27-s − 5.19·28-s − 1.90·29-s + 0.0423·30-s + ⋯ |
| L(s) = 1 | − 0.416·2-s + 0.0484·3-s − 0.826·4-s − 0.383·5-s − 0.0201·6-s + 1.18·7-s + 0.761·8-s − 0.997·9-s + 0.159·10-s − 0.0400·12-s − 0.405·13-s − 0.494·14-s − 0.0185·15-s + 0.508·16-s − 1.21·17-s + 0.415·18-s + 0.162·19-s + 0.316·20-s + 0.0575·21-s + 1.52·23-s + 0.0368·24-s − 0.853·25-s + 0.169·26-s − 0.0967·27-s − 0.980·28-s − 0.354·29-s + 0.00774·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8987477675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8987477675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.589T + 2T^{2} \) |
| 3 | \( 1 - 0.0839T + 3T^{2} \) |
| 5 | \( 1 + 0.856T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 0.707T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 - 0.843T + 31T^{2} \) |
| 37 | \( 1 - 0.681T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 0.116T + 43T^{2} \) |
| 47 | \( 1 - 0.0286T + 47T^{2} \) |
| 53 | \( 1 + 0.375T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 67 | \( 1 - 7.45T + 67T^{2} \) |
| 71 | \( 1 + 2.51T + 71T^{2} \) |
| 73 | \( 1 + 0.865T + 73T^{2} \) |
| 79 | \( 1 - 3.45T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.097264755981395855665173047675, −7.45895793964958516008444157707, −6.65901357798482596359137570872, −5.56629994357559476119598037641, −5.02737930382136757228344234563, −4.45482197797288894509095838182, −3.65581971251724818347281349797, −2.62282279868203446449255098505, −1.66271991522559424643557516938, −0.51762340029912080905468254591,
0.51762340029912080905468254591, 1.66271991522559424643557516938, 2.62282279868203446449255098505, 3.65581971251724818347281349797, 4.45482197797288894509095838182, 5.02737930382136757228344234563, 5.56629994357559476119598037641, 6.65901357798482596359137570872, 7.45895793964958516008444157707, 8.097264755981395855665173047675
