| L(s) = 1 | − 2·7-s + 30·11-s + 4·13-s − 90·17-s − 28·19-s − 120·23-s + 210·29-s − 4·31-s − 200·37-s + 240·41-s + 136·43-s + 120·47-s − 339·49-s + 30·53-s − 450·59-s − 166·61-s − 908·67-s − 1.02e3·71-s + 250·73-s − 60·77-s − 916·79-s + 1.14e3·83-s − 420·89-s − 8·91-s − 1.53e3·97-s + 450·101-s + 1.15e3·103-s + ⋯ |
| L(s) = 1 | − 0.107·7-s + 0.822·11-s + 0.0853·13-s − 1.28·17-s − 0.338·19-s − 1.08·23-s + 1.34·29-s − 0.0231·31-s − 0.888·37-s + 0.914·41-s + 0.482·43-s + 0.372·47-s − 0.988·49-s + 0.0777·53-s − 0.992·59-s − 0.348·61-s − 1.65·67-s − 1.70·71-s + 0.400·73-s − 0.0888·77-s − 1.30·79-s + 1.50·83-s − 0.500·89-s − 0.00921·91-s − 1.60·97-s + 0.443·101-s + 1.10·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 T + p^{3} T^{2} \) |
| 37 | \( 1 + 200 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 136 T + p^{3} T^{2} \) |
| 47 | \( 1 - 120 T + p^{3} T^{2} \) |
| 53 | \( 1 - 30 T + p^{3} T^{2} \) |
| 59 | \( 1 + 450 T + p^{3} T^{2} \) |
| 61 | \( 1 + 166 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 916 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1140 T + p^{3} T^{2} \) |
| 89 | \( 1 + 420 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1538 T + p^{3} T^{2} \) |
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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
−9.183281718334979095941926348086, −8.619551473087558809135885954236, −7.60042174260386014822383649991, −6.59368094075629120129597767909, −6.03158287429887492092048419911, −4.68494420080731434163410779334, −3.96640285146288481612943507712, −2.69487734458965636240521393163, −1.50558495649522183573116609513, 0,
1.50558495649522183573116609513, 2.69487734458965636240521393163, 3.96640285146288481612943507712, 4.68494420080731434163410779334, 6.03158287429887492092048419911, 6.59368094075629120129597767909, 7.60042174260386014822383649991, 8.619551473087558809135885954236, 9.183281718334979095941926348086
