| L(s) = 1 | − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s − 4·19-s + 23-s − 25-s − 2·29-s − 8·31-s − 2·35-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s + 8·55-s + 4·59-s + 10·61-s − 4·65-s − 4·67-s + 8·71-s − 6·73-s − 4·77-s − 12·83-s − 12·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 0.702·73-s − 0.455·77-s − 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351560335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351560335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.88835982643853, −13.17269082915420, −12.88671866512394, −12.35916997285268, −11.88790861532318, −11.36140684242406, −10.79344006714452, −10.56818417713680, −9.974834205889178, −9.309262519863630, −8.648522175060916, −8.339557706263207, −7.704065353831256, −7.420504834904778, −6.955284100520333, −5.955043265271207, −5.612387675690604, −5.158084474170793, −4.384508479602707, −3.809281206208688, −3.476483775039459, −2.615947782103817, −2.063308555442323, −1.179772394524797, −0.3973652993486306,
0.3973652993486306, 1.179772394524797, 2.063308555442323, 2.615947782103817, 3.476483775039459, 3.809281206208688, 4.384508479602707, 5.158084474170793, 5.612387675690604, 5.955043265271207, 6.955284100520333, 7.420504834904778, 7.704065353831256, 8.339557706263207, 8.648522175060916, 9.309262519863630, 9.974834205889178, 10.56818417713680, 10.79344006714452, 11.36140684242406, 11.88790861532318, 12.35916997285268, 12.88671866512394, 13.17269082915420, 13.88835982643853
