L(s) = 1 | − 5-s − 2·11-s + 4·13-s − 6·17-s + 8·19-s − 23-s + 25-s − 4·29-s − 2·37-s + 2·41-s − 2·43-s − 12·47-s − 7·49-s + 6·53-s + 2·55-s + 8·59-s + 2·61-s − 4·65-s + 6·67-s + 10·71-s − 2·73-s + 8·79-s + 8·83-s + 6·85-s + 12·89-s − 8·95-s − 16·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.603·11-s + 1.10·13-s − 1.45·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.742·29-s − 0.328·37-s + 0.312·41-s − 0.304·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.269·55-s + 1.04·59-s + 0.256·61-s − 0.496·65-s + 0.733·67-s + 1.18·71-s − 0.234·73-s + 0.900·79-s + 0.878·83-s + 0.650·85-s + 1.27·89-s − 0.820·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.00238988159460, −15.90993875531305, −15.08650195446301, −14.69420880024033, −13.74390637857118, −13.50764473621153, −12.98944160789308, −12.29875744062695, −11.50877543883034, −11.29535388763713, −10.71549663525466, −9.965499579193386, −9.351054373931219, −8.821079723746681, −7.987499014143275, −7.833959318130954, −6.709033670476802, −6.598320544575765, −5.381364933584454, −5.206338447600375, −4.160565465716876, −3.622881404362694, −2.914111077012869, −2.012355125800027, −1.083206569290389, 0,
1.083206569290389, 2.012355125800027, 2.914111077012869, 3.622881404362694, 4.160565465716876, 5.206338447600375, 5.381364933584454, 6.598320544575765, 6.709033670476802, 7.833959318130954, 7.987499014143275, 8.821079723746681, 9.351054373931219, 9.965499579193386, 10.71549663525466, 11.29535388763713, 11.50877543883034, 12.29875744062695, 12.98944160789308, 13.50764473621153, 13.74390637857118, 14.69420880024033, 15.08650195446301, 15.90993875531305, 16.00238988159460