| L(s) = 1 | − 4·7-s − 2·11-s + 13-s − 6·17-s + 4·19-s − 4·23-s − 10·29-s + 8·31-s + 2·37-s − 4·43-s − 2·47-s + 9·49-s − 2·53-s + 10·59-s + 10·61-s + 8·67-s + 2·71-s + 10·73-s + 8·77-s − 8·79-s − 6·83-s + 12·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s − 1.85·29-s + 1.43·31-s + 0.328·37-s − 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.274·53-s + 1.30·59-s + 1.28·61-s + 0.977·67-s + 0.237·71-s + 1.17·73-s + 0.911·77-s − 0.900·79-s − 0.658·83-s + 1.27·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.92510725225134, −14.31764830285277, −13.57956290813401, −13.23062030509142, −13.05957606459966, −12.34039964851241, −11.75394337276787, −11.24585113831891, −10.72415428258314, −9.978350152987954, −9.704686140427164, −9.263835737661182, −8.501257767453184, −8.075167726604508, −7.321011956423130, −6.743992448766432, −6.407989106078661, −5.687718429883635, −5.235951665020738, −4.345552561455069, −3.786727840035262, −3.215887440863487, −2.521650309440568, −1.945776512239090, −0.7458440316785662, 0,
0.7458440316785662, 1.945776512239090, 2.521650309440568, 3.215887440863487, 3.786727840035262, 4.345552561455069, 5.235951665020738, 5.687718429883635, 6.407989106078661, 6.743992448766432, 7.321011956423130, 8.075167726604508, 8.501257767453184, 9.263835737661182, 9.704686140427164, 9.978350152987954, 10.72415428258314, 11.24585113831891, 11.75394337276787, 12.34039964851241, 13.05957606459966, 13.23062030509142, 13.57956290813401, 14.31764830285277, 14.92510725225134
