| L(s) = 1 | − 2.79·5-s − 3.28·7-s + 11-s + 13-s − 5.66·17-s − 4.31·19-s + 2.86·23-s + 2.82·25-s + 3.51·29-s − 6.52·31-s + 9.17·35-s − 6.14·37-s − 5.28·41-s − 3.66·43-s − 2.41·47-s + 3.76·49-s + 1.03·53-s − 2.79·55-s + 1.30·59-s − 6.41·61-s − 2.79·65-s − 2.10·67-s + 7.38·71-s + 13.9·73-s − 3.28·77-s + 2.06·79-s + 0.272·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 1.23·7-s + 0.301·11-s + 0.277·13-s − 1.37·17-s − 0.990·19-s + 0.597·23-s + 0.565·25-s + 0.653·29-s − 1.17·31-s + 1.55·35-s − 1.01·37-s − 0.824·41-s − 0.558·43-s − 0.352·47-s + 0.537·49-s + 0.142·53-s − 0.377·55-s + 0.170·59-s − 0.821·61-s − 0.347·65-s − 0.257·67-s + 0.876·71-s + 1.62·73-s − 0.373·77-s + 0.232·79-s + 0.0299·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5347441625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5347441625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 - 7.38T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 - 0.272T + 83T^{2} \) |
| 89 | \( 1 + 4.03T + 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−8.350773876896344694974873558484, −7.40365229664817268397017123497, −6.66991423485122003941230129263, −6.42573791147703179155464578557, −5.21169491938514134718742113486, −4.34281274596217632318541755283, −3.70154947675809925674371753809, −3.09402051849803914865247965296, −1.94487782654726107669283207174, −0.37731140850979540634718841108,
0.37731140850979540634718841108, 1.94487782654726107669283207174, 3.09402051849803914865247965296, 3.70154947675809925674371753809, 4.34281274596217632318541755283, 5.21169491938514134718742113486, 6.42573791147703179155464578557, 6.66991423485122003941230129263, 7.40365229664817268397017123497, 8.350773876896344694974873558484
