| L(s) = 1 | + 14.2·2-s − 145.·3-s − 309.·4-s − 625·5-s − 2.06e3·6-s − 1.16e4·8-s + 1.48e3·9-s − 8.88e3·10-s + 3.04e3·11-s + 4.50e4·12-s + 6.70e4·13-s + 9.09e4·15-s − 7.52e3·16-s − 2.62e5·17-s + 2.10e4·18-s − 5.12e4·19-s + 1.93e5·20-s + 4.33e4·22-s − 8.29e5·23-s + 1.70e6·24-s + 3.90e5·25-s + 9.53e5·26-s + 2.64e6·27-s + 7.28e6·29-s + 1.29e6·30-s − 2.35e5·31-s + 5.87e6·32-s + ⋯ |
| L(s) = 1 | + 0.628·2-s − 1.03·3-s − 0.605·4-s − 0.447·5-s − 0.651·6-s − 1.00·8-s + 0.0752·9-s − 0.281·10-s + 0.0627·11-s + 0.627·12-s + 0.650·13-s + 0.463·15-s − 0.0287·16-s − 0.762·17-s + 0.0473·18-s − 0.0901·19-s + 0.270·20-s + 0.0394·22-s − 0.617·23-s + 1.04·24-s + 0.200·25-s + 0.409·26-s + 0.958·27-s + 1.91·29-s + 0.291·30-s − 0.0458·31-s + 0.990·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 14.2T + 512T^{2} \) |
| 3 | \( 1 + 145.T + 1.96e4T^{2} \) |
| 11 | \( 1 - 3.04e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.70e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.62e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.12e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.29e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.35e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.69e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.53e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.23e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.45e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.88e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.73e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.30e5T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.97e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.53e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.19e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.71e6T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.13e7T + 7.60e17T^{2} \) |
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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
−10.21635936526294470118110433808, −8.964951284949996475433056360474, −8.128412003468923133807757991013, −6.56691315593058990094368663923, −5.90199772295635006821270743922, −4.82094496262631651118226866602, −4.10721213241235676007076475337, −2.83374394224678475896757310621, −0.933840919838091740320369089580, 0,
0.933840919838091740320369089580, 2.83374394224678475896757310621, 4.10721213241235676007076475337, 4.82094496262631651118226866602, 5.90199772295635006821270743922, 6.56691315593058990094368663923, 8.128412003468923133807757991013, 8.964951284949996475433056360474, 10.21635936526294470118110433808
