| L(s) = 1 | − 32·2-s − 236·3-s + 1.02e3·4-s − 3.12e3·5-s + 7.55e3·6-s − 3.33e4·7-s − 3.27e4·8-s − 3.35e3·9-s + 1.00e5·10-s − 2.41e5·12-s + 1.06e6·14-s + 7.37e5·15-s + 1.04e6·16-s + 1.07e5·18-s − 3.20e6·20-s + 7.87e6·21-s + 1.16e6·23-s + 7.73e6·24-s + 9.76e6·25-s + 1.47e7·27-s − 3.41e7·28-s − 3.81e7·29-s − 2.36e7·30-s − 3.35e7·32-s + 1.04e8·35-s − 3.43e6·36-s + 1.02e8·40-s + ⋯ |
| L(s) = 1 | − 2-s − 0.971·3-s + 4-s − 5-s + 0.971·6-s − 1.98·7-s − 8-s − 0.0567·9-s + 10-s − 0.971·12-s + 1.98·14-s + 0.971·15-s + 16-s + 0.0567·18-s − 20-s + 1.92·21-s + 0.181·23-s + 0.971·24-s + 25-s + 1.02·27-s − 1.98·28-s − 1.86·29-s − 0.971·30-s − 32-s + 1.98·35-s − 0.0567·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.1784159426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1784159426\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 3 | \( 1 + 236 T + p^{10} T^{2} \) |
| 7 | \( 1 + 33364 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( 1 - 1169564 T + p^{10} T^{2} \) |
| 29 | \( 1 + 38179702 T + p^{10} T^{2} \) |
| 31 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( 1 + 211028098 T + p^{10} T^{2} \) |
| 43 | \( 1 - 223663364 T + p^{10} T^{2} \) |
| 47 | \( 1 + 96887764 T + p^{10} T^{2} \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 1041591898 T + p^{10} T^{2} \) |
| 67 | \( 1 + 2343243964 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 83 | \( 1 + 5449159036 T + p^{10} T^{2} \) |
| 89 | \( 1 - 11118190898 T + p^{10} T^{2} \) |
| 97 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
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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
−16.31684264289544593287696268868, −15.32351672051079809947043258019, −12.73139764353109946293858398696, −11.69292182417014220521314818307, −10.47763374456287487335340679971, −9.074917975786792076033553722316, −7.21362225182099016735773539887, −6.03734434051239795417540795063, −3.26780968355619251692508410097, −0.36640035419485094484829045350,
0.36640035419485094484829045350, 3.26780968355619251692508410097, 6.03734434051239795417540795063, 7.21362225182099016735773539887, 9.074917975786792076033553722316, 10.47763374456287487335340679971, 11.69292182417014220521314818307, 12.73139764353109946293858398696, 15.32351672051079809947043258019, 16.31684264289544593287696268868
