| L(s) = 1 | − 216.·3-s − 625·5-s − 1.66e3·7-s + 2.73e4·9-s − 3.43e4·11-s + 1.62e5·13-s + 1.35e5·15-s + 9.17e4·17-s + 5.36e5·19-s + 3.60e5·21-s − 2.34e5·23-s + 3.90e5·25-s − 1.65e6·27-s + 7.25e6·29-s − 8.23e6·31-s + 7.45e6·33-s + 1.03e6·35-s − 1.56e7·37-s − 3.52e7·39-s − 1.67e7·41-s − 9.36e6·43-s − 1.70e7·45-s + 2.69e7·47-s − 3.75e7·49-s − 1.98e7·51-s + 3.41e7·53-s + 2.14e7·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s − 0.447·5-s − 0.261·7-s + 1.38·9-s − 0.707·11-s + 1.57·13-s + 0.691·15-s + 0.266·17-s + 0.943·19-s + 0.404·21-s − 0.175·23-s + 0.200·25-s − 0.598·27-s + 1.90·29-s − 1.60·31-s + 1.09·33-s + 0.117·35-s − 1.37·37-s − 2.44·39-s − 0.924·41-s − 0.417·43-s − 0.620·45-s + 0.805·47-s − 0.931·49-s − 0.411·51-s + 0.593·53-s + 0.316·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| good | 3 | \( 1 + 216.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 1.66e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.43e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.62e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 9.17e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.36e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.34e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.23e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.67e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.36e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.41e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.12e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.49e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.58e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.53e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.74e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.14e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.25e9T + 7.60e17T^{2} \) |
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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
−11.85592749226455355071804265005, −10.99116435908604009369657361244, −10.17404396224519152151585015490, −8.501387941771958036599842709686, −7.06497412550734060731152968633, −5.96450799659414777147835591559, −4.99426723584683477078153551061, −3.47194786033776950879814593948, −1.18736901032669014144868414064, 0,
1.18736901032669014144868414064, 3.47194786033776950879814593948, 4.99426723584683477078153551061, 5.96450799659414777147835591559, 7.06497412550734060731152968633, 8.501387941771958036599842709686, 10.17404396224519152151585015490, 10.99116435908604009369657361244, 11.85592749226455355071804265005
