| L(s) = 1 | − 3-s − 2·7-s + 9-s − 13-s − 2·17-s − 2·19-s + 2·21-s − 8·23-s − 27-s + 6·29-s + 2·31-s + 6·37-s + 39-s − 4·43-s − 8·47-s − 3·49-s + 2·51-s + 6·53-s + 2·57-s + 4·59-s + 2·61-s − 2·63-s − 2·67-s + 8·69-s + 4·71-s + 2·73-s + 12·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.436·21-s − 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.986·37-s + 0.160·39-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.963·69-s + 0.474·71-s + 0.234·73-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.43261381377735, −14.86862016652658, −14.32189973310468, −13.63524572917577, −13.22315586777419, −12.69322512777468, −12.09434136078966, −11.74695247385399, −11.14462386936097, −10.36469186539355, −10.12782359084807, −9.542725189872137, −8.945575067030264, −8.139149088527018, −7.826938784250187, −6.900805140072807, −6.336683341201082, −6.218694728993037, −5.253045514472212, −4.709688646877307, −4.038794486698128, −3.426219401952989, −2.530376004952732, −1.934958098430834, −0.8182746791604373, 0,
0.8182746791604373, 1.934958098430834, 2.530376004952732, 3.426219401952989, 4.038794486698128, 4.709688646877307, 5.253045514472212, 6.218694728993037, 6.336683341201082, 6.900805140072807, 7.826938784250187, 8.139149088527018, 8.945575067030264, 9.542725189872137, 10.12782359084807, 10.36469186539355, 11.14462386936097, 11.74695247385399, 12.09434136078966, 12.69322512777468, 13.22315586777419, 13.63524572917577, 14.32189973310468, 14.86862016652658, 15.43261381377735
