| L(s) = 1 | + 11.8·2-s − 26.1·3-s + 12.1·4-s − 422.·5-s − 309.·6-s − 1.21e3·7-s − 1.37e3·8-s − 1.50e3·9-s − 4.99e3·10-s + 4.48e3·11-s − 317.·12-s − 1.43e4·14-s + 1.10e4·15-s − 1.77e4·16-s − 3.44e4·17-s − 1.77e4·18-s − 2.83e4·19-s − 5.12e3·20-s + 3.17e4·21-s + 5.30e4·22-s + 3.09e4·23-s + 3.58e4·24-s + 1.00e5·25-s + 9.64e4·27-s − 1.47e4·28-s − 4.40e4·29-s + 1.30e5·30-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.559·3-s + 0.0948·4-s − 1.51·5-s − 0.585·6-s − 1.33·7-s − 0.947·8-s − 0.687·9-s − 1.58·10-s + 1.01·11-s − 0.0530·12-s − 1.40·14-s + 0.844·15-s − 1.08·16-s − 1.69·17-s − 0.719·18-s − 0.948·19-s − 0.143·20-s + 0.749·21-s + 1.06·22-s + 0.531·23-s + 0.529·24-s + 1.28·25-s + 0.943·27-s − 0.127·28-s − 0.335·29-s + 0.883·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2488781073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2488781073\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 11.8T + 128T^{2} \) |
| 3 | \( 1 + 26.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 422.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.21e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.48e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.44e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.83e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.40e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.80e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.44e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.16e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.76e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.98e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.65e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.30e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.77e6T + 8.07e13T^{2} \) |
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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
−11.63764346999167822684200839254, −11.03580155421522753934697638696, −9.307072892708129184795018843596, −8.488492019044810953699131033525, −6.74604100756608969868510229340, −6.24079584307209648768126305645, −4.69478155351847089055341025613, −3.88640533249382652520563794263, −2.95125690535833968791628242455, −0.22743535763911488355070707700,
0.22743535763911488355070707700, 2.95125690535833968791628242455, 3.88640533249382652520563794263, 4.69478155351847089055341025613, 6.24079584307209648768126305645, 6.74604100756608969868510229340, 8.488492019044810953699131033525, 9.307072892708129184795018843596, 11.03580155421522753934697638696, 11.63764346999167822684200839254
