| L(s) = 1 | − 40.6·2-s − 254.·3-s + 1.14e3·4-s − 1.76e3·5-s + 1.03e4·6-s + 1.27e3·7-s − 2.56e4·8-s + 4.48e4·9-s + 7.16e4·10-s − 5.26e4·11-s − 2.90e5·12-s − 5.20e4·14-s + 4.47e5·15-s + 4.56e5·16-s − 2.23e5·17-s − 1.82e6·18-s − 5.45e5·19-s − 2.01e6·20-s − 3.25e5·21-s + 2.13e6·22-s − 1.10e6·23-s + 6.50e6·24-s + 1.15e6·25-s − 6.40e6·27-s + 1.46e6·28-s − 1.27e6·29-s − 1.82e7·30-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.81·3-s + 2.22·4-s − 1.26·5-s + 3.25·6-s + 0.201·7-s − 2.20·8-s + 2.28·9-s + 2.26·10-s − 1.08·11-s − 4.03·12-s − 0.361·14-s + 2.28·15-s + 1.74·16-s − 0.648·17-s − 4.09·18-s − 0.959·19-s − 2.81·20-s − 0.364·21-s + 1.94·22-s − 0.823·23-s + 4.00·24-s + 0.589·25-s − 2.31·27-s + 0.449·28-s − 0.335·29-s − 4.10·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 40.6T + 512T^{2} \) |
| 3 | \( 1 + 254.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.76e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.26e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.23e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.45e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.10e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.27e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.41e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.37e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.62e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.73e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.01e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.03e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.26e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.27e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.31e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.88e8T + 7.60e17T^{2} \) |
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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
−10.89414842257844439961049681599, −9.830447285552277227132233124694, −8.412009732832923828051355973849, −7.54631147620291118977261153187, −6.77591111575741351450941866386, −5.57897680379157977825218894289, −4.17949615239517128010638071684, −2.02824436131446940760711689145, −0.59067012967632484528446641233, 0,
0.59067012967632484528446641233, 2.02824436131446940760711689145, 4.17949615239517128010638071684, 5.57897680379157977825218894289, 6.77591111575741351450941866386, 7.54631147620291118977261153187, 8.412009732832923828051355973849, 9.830447285552277227132233124694, 10.89414842257844439961049681599
