| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 2·5-s + 2·6-s + 9-s − 4·10-s + 11-s + 2·12-s − 13-s − 2·15-s − 4·16-s − 17-s + 2·18-s + 2·19-s − 4·20-s + 2·22-s + 4·23-s − 25-s − 2·26-s + 27-s − 4·30-s + 5·31-s − 8·32-s + 33-s − 2·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.577·12-s − 0.277·13-s − 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s + 0.458·19-s − 0.894·20-s + 0.426·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.730·30-s + 0.898·31-s − 1.41·32-s + 0.174·33-s − 0.342·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.01151731172872, −14.77183395862939, −14.34540474012948, −13.63864059213012, −13.27685471950268, −12.82692540589163, −12.14847988247340, −11.80031020884157, −11.31191569269760, −10.78110229138573, −9.842355446143174, −9.439973144607756, −8.737832881036241, −8.163236152288518, −7.588644105867894, −6.991590917877555, −6.436539237064529, −5.818458037263651, −4.971963901053899, −4.579371335063579, −4.039460711392025, −3.355744779870100, −2.970969488757222, −2.221282388396488, −1.200670192476957, 0,
1.200670192476957, 2.221282388396488, 2.970969488757222, 3.355744779870100, 4.039460711392025, 4.579371335063579, 4.971963901053899, 5.818458037263651, 6.436539237064529, 6.991590917877555, 7.588644105867894, 8.163236152288518, 8.737832881036241, 9.439973144607756, 9.842355446143174, 10.78110229138573, 11.31191569269760, 11.80031020884157, 12.14847988247340, 12.82692540589163, 13.27685471950268, 13.63864059213012, 14.34540474012948, 14.77183395862939, 15.01151731172872
