L(s) = 1 | − 43.7·2-s − 83.5·3-s + 1.40e3·4-s + 712.·5-s + 3.65e3·6-s − 5.54e3·7-s − 3.89e4·8-s − 1.26e4·9-s − 3.11e4·10-s − 3.60e4·11-s − 1.17e5·12-s − 1.75e5·13-s + 2.42e5·14-s − 5.95e4·15-s + 9.84e5·16-s + 5.55e5·18-s − 5.12e5·19-s + 9.98e5·20-s + 4.63e5·21-s + 1.57e6·22-s + 1.41e6·23-s + 3.25e6·24-s − 1.44e6·25-s + 7.67e6·26-s + 2.70e6·27-s − 7.76e6·28-s + 6.23e6·29-s + ⋯ |
L(s) = 1 | − 1.93·2-s − 0.595·3-s + 2.73·4-s + 0.509·5-s + 1.15·6-s − 0.872·7-s − 3.35·8-s − 0.644·9-s − 0.985·10-s − 0.743·11-s − 1.63·12-s − 1.70·13-s + 1.68·14-s − 0.303·15-s + 3.75·16-s + 1.24·18-s − 0.902·19-s + 1.39·20-s + 0.519·21-s + 1.43·22-s + 1.05·23-s + 2.00·24-s − 0.740·25-s + 3.29·26-s + 0.980·27-s − 2.38·28-s + 1.63·29-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 | \( 1 + 43.7T + 512T^{2} \) |
| 3 | \( 1 + 83.5T + 1.96e4T^{2} \) |
| 5 | \( 1 - 712.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.54e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.75e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.00e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.77e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.78e5T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.35e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.00e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.95e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.53e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.41e9T + 7.60e17T^{2} \) |
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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
−9.960593838824474536379847879837, −8.926846419625656182062959434182, −8.028379750971990761718536638914, −6.94181220028482448644306614783, −6.29064451850218595286568807680, −5.20806478353818132942264414599, −2.83757287330653248847249689010, −2.29472130319417651814305138926, −0.72308549512638738047794863888, 0,
0.72308549512638738047794863888, 2.29472130319417651814305138926, 2.83757287330653248847249689010, 5.20806478353818132942264414599, 6.29064451850218595286568807680, 6.94181220028482448644306614783, 8.028379750971990761718536638914, 8.926846419625656182062959434182, 9.960593838824474536379847879837