| L(s) = 1 | − 3-s − 5-s + 9-s + 11-s + 2·13-s + 15-s − 8·19-s − 7·23-s − 4·25-s − 27-s − 31-s − 33-s + 6·37-s − 2·39-s + 6·43-s − 45-s + 7·47-s − 7·49-s + 2·53-s − 55-s + 8·57-s − 4·59-s − 4·61-s − 2·65-s + 3·67-s + 7·69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.83·19-s − 1.45·23-s − 4/5·25-s − 0.192·27-s − 0.179·31-s − 0.174·33-s + 0.986·37-s − 0.320·39-s + 0.914·43-s − 0.149·45-s + 1.02·47-s − 49-s + 0.274·53-s − 0.134·55-s + 1.05·57-s − 0.520·59-s − 0.512·61-s − 0.248·65-s + 0.366·67-s + 0.842·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.58941717140487, −12.90107967606405, −12.61955149903271, −12.13275936389505, −11.56302725245729, −11.31462722930735, −10.63952651557867, −10.42479651931996, −9.765017803033048, −9.253918032398601, −8.705812730760720, −8.176360270438860, −7.764888120127740, −7.268468484234019, −6.474521888740088, −6.206658573406513, −5.836510899861749, −5.124341305825149, −4.324266351695860, −4.133995904179570, −3.684391525765300, −2.773474323743266, −2.096057243188767, −1.583310048352343, −0.6630519564924271, 0,
0.6630519564924271, 1.583310048352343, 2.096057243188767, 2.773474323743266, 3.684391525765300, 4.133995904179570, 4.324266351695860, 5.124341305825149, 5.836510899861749, 6.206658573406513, 6.474521888740088, 7.268468484234019, 7.764888120127740, 8.176360270438860, 8.705812730760720, 9.253918032398601, 9.765017803033048, 10.42479651931996, 10.63952651557867, 11.31462722930735, 11.56302725245729, 12.13275936389505, 12.61955149903271, 12.90107967606405, 13.58941717140487
