| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s − 11-s + 4·13-s − 15-s + 17-s + 4·19-s − 3·21-s − 4·23-s + 25-s + 27-s − 29-s + 4·31-s − 33-s + 3·35-s + 2·37-s + 4·39-s − 7·41-s − 2·43-s − 45-s − 7·47-s + 2·49-s + 51-s + 11·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.718·31-s − 0.174·33-s + 0.507·35-s + 0.328·37-s + 0.640·39-s − 1.09·41-s − 0.304·43-s − 0.149·45-s − 1.02·47-s + 2/7·49-s + 0.140·51-s + 1.51·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−13.41666095671890, −13.10721475102908, −12.43762614647533, −12.08610802000953, −11.49476906222466, −11.16918838604368, −10.24993773901976, −10.19701821366717, −9.599835018921550, −9.153445677173192, −8.563720222093690, −8.102898207425849, −7.824247273361277, −7.060778189560073, −6.623430356764522, −6.285937044340694, −5.457927813633709, −5.208710989981123, −4.187895658536573, −3.904182046698732, −3.308463527056713, −2.976184110009126, −2.288582110053830, −1.483768249087926, −0.8223761693481491, 0,
0.8223761693481491, 1.483768249087926, 2.288582110053830, 2.976184110009126, 3.308463527056713, 3.904182046698732, 4.187895658536573, 5.208710989981123, 5.457927813633709, 6.285937044340694, 6.623430356764522, 7.060778189560073, 7.824247273361277, 8.102898207425849, 8.563720222093690, 9.153445677173192, 9.599835018921550, 10.19701821366717, 10.24993773901976, 11.16918838604368, 11.49476906222466, 12.08610802000953, 12.43762614647533, 13.10721475102908, 13.41666095671890
