| L(s) = 1 | + 0.797·2-s + 3-s − 1.36·4-s + 0.229·5-s + 0.797·6-s − 7-s − 2.68·8-s + 9-s + 0.183·10-s − 2.28·11-s − 1.36·12-s − 2.54·13-s − 0.797·14-s + 0.229·15-s + 0.585·16-s − 5.30·17-s + 0.797·18-s + 3.06·19-s − 0.313·20-s − 21-s − 1.82·22-s + 4.59·23-s − 2.68·24-s − 4.94·25-s − 2.03·26-s + 27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 0.564·2-s + 0.577·3-s − 0.681·4-s + 0.102·5-s + 0.325·6-s − 0.377·7-s − 0.948·8-s + 0.333·9-s + 0.0580·10-s − 0.687·11-s − 0.393·12-s − 0.707·13-s − 0.213·14-s + 0.0593·15-s + 0.146·16-s − 1.28·17-s + 0.188·18-s + 0.703·19-s − 0.0701·20-s − 0.218·21-s − 0.388·22-s + 0.958·23-s − 0.547·24-s − 0.989·25-s − 0.398·26-s + 0.192·27-s + 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998610052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998610052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 0.797T + 2T^{2} \) |
| 5 | \( 1 - 0.229T + 5T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.36T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.500337029033019140238728008304, −7.78486110430875869515473554768, −6.96157526489909059238301334116, −6.16145415988851039275522861019, −5.27396568283989103121847926458, −4.63127362021324119349136046216, −3.94194461907649708798472443971, −2.91231247232290987795640207550, −2.41559876771941025236864321671, −0.70662498188415935409855075327,
0.70662498188415935409855075327, 2.41559876771941025236864321671, 2.91231247232290987795640207550, 3.94194461907649708798472443971, 4.63127362021324119349136046216, 5.27396568283989103121847926458, 6.16145415988851039275522861019, 6.96157526489909059238301334116, 7.78486110430875869515473554768, 8.500337029033019140238728008304
