| L(s) = 1 | + 3·2-s − 18·3-s − 41·4-s + 50·5-s − 54·6-s + 98·7-s − 177·8-s + 243·9-s + 150·10-s − 232·11-s + 738·12-s − 204·13-s + 294·14-s − 900·15-s + 783·16-s − 932·17-s + 729·18-s − 2.55e3·19-s − 2.05e3·20-s − 1.76e3·21-s − 696·22-s − 6.92e3·23-s + 3.18e3·24-s + 1.87e3·25-s − 612·26-s − 2.91e3·27-s − 4.01e3·28-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 1.15·3-s − 1.28·4-s + 0.894·5-s − 0.612·6-s + 0.755·7-s − 0.977·8-s + 9-s + 0.474·10-s − 0.578·11-s + 1.47·12-s − 0.334·13-s + 0.400·14-s − 1.03·15-s + 0.764·16-s − 0.782·17-s + 0.530·18-s − 1.62·19-s − 1.14·20-s − 0.872·21-s − 0.306·22-s − 2.72·23-s + 1.12·24-s + 3/5·25-s − 0.177·26-s − 0.769·27-s − 0.968·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 25 p T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 232 T + 331398 T^{2} + 232 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 204 T - 121650 T^{2} + 204 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 932 T + 3047510 T^{2} + 932 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2552 T + 5877334 T^{2} + 2552 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6920 T + 24610286 T^{2} + 6920 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 916 T + 6403502 T^{2} - 916 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7080 T + 56510142 T^{2} + 7080 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10988 T + 85579390 T^{2} + 10988 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 964 T + 198248726 T^{2} - 964 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6912 T + 272638182 T^{2} - 6912 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1240 T + 456577374 T^{2} - 1240 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7748 T + 830428302 T^{2} + 7748 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 36264 T + 1588873382 T^{2} - 36264 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 39244 T + 1633462446 T^{2} + 39244 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 62192 T + 3632382870 T^{2} + 62192 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10840 T - 1427847858 T^{2} - 10840 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 19388 T + 3699482182 T^{2} - 19388 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 39952 T + 5758722334 T^{2} + 39952 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 30376 T + 980675670 T^{2} - 30376 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 154372 T + 15005046134 T^{2} - 154372 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 58564 T + 6617108278 T^{2} + 58564 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41107311161863469491061752375, −12.37117115918208198809773511203, −11.60443377819367680166704220152, −10.87968549499940579447649237765, −10.30341541991000937958339095716, −10.22262519390159328178012967070, −9.210679780824707420760907195739, −8.933538109043725441055080863807, −8.140249446974244518548307600530, −7.58214847735602730620435913410, −6.49796425919086915218297286510, −6.13521947995429647310246133261, −5.27812380063715445804326295259, −5.18117315961642254666546545501, −4.18795889079453044501671104183, −4.03430118428888771360269381573, −2.32300834676924795477934388441, −1.63328601963624453741972388368, 0, 0,
1.63328601963624453741972388368, 2.32300834676924795477934388441, 4.03430118428888771360269381573, 4.18795889079453044501671104183, 5.18117315961642254666546545501, 5.27812380063715445804326295259, 6.13521947995429647310246133261, 6.49796425919086915218297286510, 7.58214847735602730620435913410, 8.140249446974244518548307600530, 8.933538109043725441055080863807, 9.210679780824707420760907195739, 10.22262519390159328178012967070, 10.30341541991000937958339095716, 10.87968549499940579447649237765, 11.60443377819367680166704220152, 12.37117115918208198809773511203, 12.41107311161863469491061752375
