| L(s) = 1 | − 2.07·2-s + 2.29·4-s − 2.34·5-s − 3.67·7-s − 0.608·8-s + 4.86·10-s − 4.28·11-s + 2.08·13-s + 7.61·14-s − 3.32·16-s + 2.12·17-s − 0.620·19-s − 5.38·20-s + 8.87·22-s − 3.35·23-s + 0.512·25-s − 4.32·26-s − 8.42·28-s − 1.37·29-s + 8.10·32-s − 4.40·34-s + 8.62·35-s − 0.274·37-s + 1.28·38-s + 1.42·40-s + 4.27·41-s − 0.269·43-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.14·4-s − 1.05·5-s − 1.38·7-s − 0.215·8-s + 1.53·10-s − 1.29·11-s + 0.578·13-s + 2.03·14-s − 0.831·16-s + 0.516·17-s − 0.142·19-s − 1.20·20-s + 1.89·22-s − 0.699·23-s + 0.102·25-s − 0.848·26-s − 1.59·28-s − 0.255·29-s + 1.43·32-s − 0.756·34-s + 1.45·35-s − 0.0451·37-s + 0.208·38-s + 0.225·40-s + 0.668·41-s − 0.0411·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 + 0.620T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 37 | \( 1 + 0.274T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 0.269T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 1.33T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 - 6.71T + 97T^{2} \) |
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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
−7.60949911511004605003355820223, −7.09285255995148136810219694774, −6.28558963185996365372866506363, −5.58533252264907300336452521525, −4.48486054455794939872980056042, −3.67258010471007838328481987362, −2.98210175680924017921901689231, −2.05661032165282278450344273628, −0.72147532087614770099858647947, 0,
0.72147532087614770099858647947, 2.05661032165282278450344273628, 2.98210175680924017921901689231, 3.67258010471007838328481987362, 4.48486054455794939872980056042, 5.58533252264907300336452521525, 6.28558963185996365372866506363, 7.09285255995148136810219694774, 7.60949911511004605003355820223
