| L(s) = 1 | + 0.933·2-s + 0.244·3-s − 1.12·4-s − 1.38·5-s + 0.227·6-s + 4.47·7-s − 2.91·8-s − 2.94·9-s − 1.28·10-s − 0.275·12-s − 3.28·13-s + 4.17·14-s − 0.337·15-s − 0.465·16-s + 4.80·17-s − 2.74·18-s − 4.66·19-s + 1.55·20-s + 1.09·21-s − 5.54·23-s − 0.712·24-s − 3.09·25-s − 3.06·26-s − 1.45·27-s − 5.05·28-s − 0.148·29-s − 0.314·30-s + ⋯ |
| L(s) = 1 | + 0.659·2-s + 0.140·3-s − 0.564·4-s − 0.617·5-s + 0.0930·6-s + 1.69·7-s − 1.03·8-s − 0.980·9-s − 0.407·10-s − 0.0796·12-s − 0.910·13-s + 1.11·14-s − 0.0870·15-s − 0.116·16-s + 1.16·17-s − 0.646·18-s − 1.07·19-s + 0.348·20-s + 0.238·21-s − 1.15·23-s − 0.145·24-s − 0.619·25-s − 0.600·26-s − 0.279·27-s − 0.955·28-s − 0.0275·29-s − 0.0574·30-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
| \(L(1)\) |
\(\approx\) |
\(1.787473867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787473867\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.933T + 2T^{2} \) |
| 3 | \( 1 - 0.244T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 + 0.148T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 - 7.71T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 - 4.86T + 59T^{2} \) |
| 67 | \( 1 - 0.456T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 0.677T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 2.17T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.79215067838746462609857347309, −7.61760070043591905478961003398, −6.18689000343814397298738054617, −5.66266294387900176601606374603, −4.94045856984073037695221474023, −4.38917430364875052929098265456, −3.79270118708278484458911314337, −2.80108090951775109683758069422, −1.99021644154947391555381061469, −0.59283884948557160354537684832,
0.59283884948557160354537684832, 1.99021644154947391555381061469, 2.80108090951775109683758069422, 3.79270118708278484458911314337, 4.38917430364875052929098265456, 4.94045856984073037695221474023, 5.66266294387900176601606374603, 6.18689000343814397298738054617, 7.61760070043591905478961003398, 7.79215067838746462609857347309
