| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 4·15-s + 4·17-s − 4·19-s + 4·23-s − 25-s − 4·27-s − 2·29-s − 2·31-s − 6·37-s + 4·41-s − 4·43-s − 2·45-s + 2·47-s + 8·51-s + 2·53-s − 8·57-s − 6·59-s + 4·61-s + 8·69-s + 12·71-s + 16·73-s − 2·75-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.03·15-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s − 0.359·31-s − 0.986·37-s + 0.624·41-s − 0.609·43-s − 0.298·45-s + 0.291·47-s + 1.12·51-s + 0.274·53-s − 1.05·57-s − 0.781·59-s + 0.512·61-s + 0.963·69-s + 1.42·71-s + 1.87·73-s − 0.230·75-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.98074875236477, −13.84537739271090, −12.92249150748811, −12.76755855046113, −12.14163839832157, −11.61845016325461, −11.09647509014473, −10.67080237783698, −9.975348728288969, −9.505719804130768, −8.978019663677440, −8.468042845662634, −8.148176061245014, −7.567110316256759, −7.245921362874297, −6.565866316572581, −5.880555951788639, −5.252193959379295, −4.663798682972945, −3.870498290133543, −3.608037853792509, −3.096090105220413, −2.350743518978191, −1.828288191242938, −0.8880900962247709, 0,
0.8880900962247709, 1.828288191242938, 2.350743518978191, 3.096090105220413, 3.608037853792509, 3.870498290133543, 4.663798682972945, 5.252193959379295, 5.880555951788639, 6.565866316572581, 7.245921362874297, 7.567110316256759, 8.148176061245014, 8.468042845662634, 8.978019663677440, 9.505719804130768, 9.975348728288969, 10.67080237783698, 11.09647509014473, 11.61845016325461, 12.14163839832157, 12.76755855046113, 12.92249150748811, 13.84537739271090, 13.98074875236477
