L(s) = 1 | + 5-s + 4·11-s − 6·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s + 6·29-s + 31-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s − 10·53-s + 4·55-s + 12·59-s + 2·61-s − 6·65-s − 4·67-s + 2·73-s − 4·83-s − 2·85-s + 14·89-s + 4·95-s + 18·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.179·31-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s − 1.37·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s − 0.744·65-s − 0.488·67-s + 0.234·73-s − 0.439·83-s − 0.216·85-s + 1.48·89-s + 0.410·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.04728066205906, −13.84096200636483, −13.14551322809005, −12.58311302499048, −12.04824235033567, −11.72396850382108, −11.41061791183021, −10.46663227577032, −9.943066402222871, −9.813129451333366, −9.255617833109210, −8.591579084498257, −8.139948248860348, −7.483449813587873, −6.985011229689337, −6.413966034909470, −6.099760485962435, −5.226504545226377, −4.817748855168067, −4.324709110993289, −3.523648650851703, −3.026468833938383, −2.150178026177427, −1.817383178621889, −0.9012485736724052, 0,
0.9012485736724052, 1.817383178621889, 2.150178026177427, 3.026468833938383, 3.523648650851703, 4.324709110993289, 4.817748855168067, 5.226504545226377, 6.099760485962435, 6.413966034909470, 6.985011229689337, 7.483449813587873, 8.139948248860348, 8.591579084498257, 9.255617833109210, 9.813129451333366, 9.943066402222871, 10.46663227577032, 11.41061791183021, 11.72396850382108, 12.04824235033567, 12.58311302499048, 13.14551322809005, 13.84096200636483, 14.04728066205906