| L(s) = 1 | − 3·9-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s − 2·29-s − 10·37-s − 6·41-s − 4·43-s − 4·47-s − 7·49-s − 2·53-s + 12·59-s + 61-s + 4·67-s − 2·73-s + 8·79-s + 9·81-s − 14·89-s − 18·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.371·29-s − 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s − 49-s − 0.274·53-s + 1.56·59-s + 0.128·61-s + 0.488·67-s − 0.234·73-s + 0.900·79-s + 81-s − 1.48·89-s − 1.82·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−15.70499302230877, −15.00217796200530, −14.58686799878234, −13.96728894027853, −13.77928982113016, −12.99381014841380, −12.32207235177579, −11.76610515786241, −11.45429930765540, −10.93243170039551, −10.07588683595152, −9.684678182925488, −8.956613573509492, −8.546554131598872, −8.022781453080550, −7.201138435288513, −6.632485035512317, −6.124090138289561, −5.325154414175716, −5.020337293570492, −3.823026636763114, −3.548481853525439, −2.826128068408654, −1.785006160720023, −1.146303226874130, 0,
1.146303226874130, 1.785006160720023, 2.826128068408654, 3.548481853525439, 3.823026636763114, 5.020337293570492, 5.325154414175716, 6.124090138289561, 6.632485035512317, 7.201138435288513, 8.022781453080550, 8.546554131598872, 8.956613573509492, 9.684678182925488, 10.07588683595152, 10.93243170039551, 11.45429930765540, 11.76610515786241, 12.32207235177579, 12.99381014841380, 13.77928982113016, 13.96728894027853, 14.58686799878234, 15.00217796200530, 15.70499302230877
