| L(s) = 1 | − 3.07·2-s + 18.6·3-s − 502.·4-s − 2.54e3·5-s − 57.2·6-s − 1.09e4·7-s + 3.11e3·8-s − 1.93e4·9-s + 7.80e3·10-s + 5.50e4·11-s − 9.35e3·12-s − 1.01e5·13-s + 3.36e4·14-s − 4.72e4·15-s + 2.47e5·16-s + 5.94e4·18-s − 2.43e5·19-s + 1.27e6·20-s − 2.03e5·21-s − 1.69e5·22-s − 7.99e5·23-s + 5.80e4·24-s + 4.49e6·25-s + 3.13e5·26-s − 7.26e5·27-s + 5.50e6·28-s + 1.08e6·29-s + ⋯ |
| L(s) = 1 | − 0.135·2-s + 0.132·3-s − 0.981·4-s − 1.81·5-s − 0.0180·6-s − 1.72·7-s + 0.269·8-s − 0.982·9-s + 0.246·10-s + 1.13·11-s − 0.130·12-s − 0.989·13-s + 0.234·14-s − 0.241·15-s + 0.944·16-s + 0.133·18-s − 0.428·19-s + 1.78·20-s − 0.228·21-s − 0.153·22-s − 0.595·23-s + 0.0357·24-s + 2.30·25-s + 0.134·26-s − 0.262·27-s + 1.69·28-s + 0.285·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 3.07T + 512T^{2} \) |
| 3 | \( 1 - 18.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.54e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.09e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.50e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.01e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 2.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.99e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.08e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.11e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.74e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.55e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.19e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.32e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.84e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 9.89e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.48e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.08e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.20e8T + 7.60e17T^{2} \) |
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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
−9.437074787992650310723401064980, −8.939914670510647004651803926072, −7.926809090072289849998915696108, −7.02581745246938906133509845072, −5.85740740253883531413032834706, −4.35044356377462712821905127414, −3.74073055574973362091398739532, −2.88570370130235083720936653788, −0.61189935441812234227697039555, 0,
0.61189935441812234227697039555, 2.88570370130235083720936653788, 3.74073055574973362091398739532, 4.35044356377462712821905127414, 5.85740740253883531413032834706, 7.02581745246938906133509845072, 7.926809090072289849998915696108, 8.939914670510647004651803926072, 9.437074787992650310723401064980
