L(s) = 1 | − 3·9-s − 6·11-s − 4·13-s − 7·17-s − 4·19-s + 23-s + 6·29-s + 3·31-s + 4·37-s − 9·41-s + 8·43-s − 11·47-s + 4·53-s − 10·59-s − 2·61-s + 8·67-s + 3·71-s − 2·73-s + 17·79-s + 9·81-s + 2·83-s − 7·89-s + 97-s + 18·99-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 9-s − 1.80·11-s − 1.10·13-s − 1.69·17-s − 0.917·19-s + 0.208·23-s + 1.11·29-s + 0.538·31-s + 0.657·37-s − 1.40·41-s + 1.21·43-s − 1.60·47-s + 0.549·53-s − 1.30·59-s − 0.256·61-s + 0.977·67-s + 0.356·71-s − 0.234·73-s + 1.91·79-s + 81-s + 0.219·83-s − 0.741·89-s + 0.101·97-s + 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−14.19505953159581, −13.72147526957911, −13.30978656834258, −12.71763594487225, −12.45459371491914, −11.68847054334714, −11.24914658393713, −10.75173590357560, −10.34876995753672, −9.824734945825509, −9.174205126254436, −8.561092184802181, −8.262863527357938, −7.717707201872697, −7.127018370591456, −6.413284878140485, −6.135061145344146, −5.258044919298143, −4.778939067863733, −4.596221565839898, −3.527436703039463, −2.797800154101559, −2.437679788844501, −1.989361832536794, −0.5805196833362792, 0,
0.5805196833362792, 1.989361832536794, 2.437679788844501, 2.797800154101559, 3.527436703039463, 4.596221565839898, 4.778939067863733, 5.258044919298143, 6.135061145344146, 6.413284878140485, 7.127018370591456, 7.717707201872697, 8.262863527357938, 8.561092184802181, 9.174205126254436, 9.824734945825509, 10.34876995753672, 10.75173590357560, 11.24914658393713, 11.68847054334714, 12.45459371491914, 12.71763594487225, 13.30978656834258, 13.72147526957911, 14.19505953159581