L(s) = 1 | − 7-s + 6·11-s − 5·13-s − 6·17-s − 5·19-s + 6·23-s + 6·29-s + 31-s − 2·37-s − 43-s − 6·47-s − 6·49-s − 12·53-s − 6·59-s − 13·61-s + 11·67-s − 2·73-s − 6·77-s − 8·79-s + 6·83-s + 5·91-s + 7·97-s + 12·101-s − 4·103-s − 12·107-s − 7·109-s + 12·113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.80·11-s − 1.38·13-s − 1.45·17-s − 1.14·19-s + 1.25·23-s + 1.11·29-s + 0.179·31-s − 0.328·37-s − 0.152·43-s − 0.875·47-s − 6/7·49-s − 1.64·53-s − 0.781·59-s − 1.66·61-s + 1.34·67-s − 0.234·73-s − 0.683·77-s − 0.900·79-s + 0.658·83-s + 0.524·91-s + 0.710·97-s + 1.19·101-s − 0.394·103-s − 1.16·107-s − 0.670·109-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.314511452179534597820674660221, −7.26879090044825103896297504040, −6.45091253684309488887986189792, −6.43227025058071167883349470739, −4.81921008362917817651377463164, −4.53076729285564080044343456884, −3.45569829363633777629454199150, −2.51682024423688394090859676004, −1.48874975899024327549158439830, 0,
1.48874975899024327549158439830, 2.51682024423688394090859676004, 3.45569829363633777629454199150, 4.53076729285564080044343456884, 4.81921008362917817651377463164, 6.43227025058071167883349470739, 6.45091253684309488887986189792, 7.26879090044825103896297504040, 8.314511452179534597820674660221