| L(s) = 1 | − 6·2-s + 7·3-s + 16·4-s − 42·6-s + 39·7-s − 24·8-s + 27·9-s − 39·11-s + 112·12-s − 26·13-s − 234·14-s + 48·16-s − 27·17-s − 162·18-s − 153·19-s + 273·21-s + 234·22-s − 57·23-s − 168·24-s + 58·25-s + 156·26-s + 224·27-s + 624·28-s + 69·29-s − 192·32-s − 273·33-s + 162·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.34·3-s + 2·4-s − 2.85·6-s + 2.10·7-s − 1.06·8-s + 9-s − 1.06·11-s + 2.69·12-s − 0.554·13-s − 4.46·14-s + 3/4·16-s − 0.385·17-s − 2.12·18-s − 1.84·19-s + 2.83·21-s + 2.26·22-s − 0.516·23-s − 1.42·24-s + 0.463·25-s + 1.17·26-s + 1.59·27-s + 4.21·28-s + 0.441·29-s − 1.06·32-s − 1.44·33-s + 0.817·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5121464513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5121464513\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 p T + 5 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 58 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 39 T + 850 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 39 T + 1838 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 27 T - 4184 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 153 T + 14662 T^{2} + 153 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 57 T - 8918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 69 T - 19628 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54290 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 69 T + 52240 T^{2} + 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 681 T + 223508 T^{2} + 681 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 85 T - 72282 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90034 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 426 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 33 T + 205742 T^{2} + 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 17 T - 226692 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 285 T + 327838 T^{2} - 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1011 T + 698618 T^{2} - 1011 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 231166 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 962026 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 531 T + 798956 T^{2} - 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2139 T + 2437780 T^{2} - 2139 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.88636005144530613870746315734, −18.85927168084700490843340412488, −18.44175080340776231630660574013, −18.07827698741142481588090997022, −17.26214822321643194606012520434, −16.99354970851497277558510659429, −15.75822802309119482862262646720, −14.96920099474815101760947911563, −14.59910339934377349528497014381, −13.78044923184104415161506893934, −12.71056452343602839582459843870, −11.57781508819803023179857827817, −10.39776468923365132375400991003, −10.30179298841582890889348698943, −8.868693986454882017884086778155, −8.360288387817286688492718603371, −8.231085512020065232240113341638, −7.18864249379947825917987191008, −4.80198786544075143735648168099, −2.11014574367122488708249141919,
2.11014574367122488708249141919, 4.80198786544075143735648168099, 7.18864249379947825917987191008, 8.231085512020065232240113341638, 8.360288387817286688492718603371, 8.868693986454882017884086778155, 10.30179298841582890889348698943, 10.39776468923365132375400991003, 11.57781508819803023179857827817, 12.71056452343602839582459843870, 13.78044923184104415161506893934, 14.59910339934377349528497014381, 14.96920099474815101760947911563, 15.75822802309119482862262646720, 16.99354970851497277558510659429, 17.26214822321643194606012520434, 18.07827698741142481588090997022, 18.44175080340776231630660574013, 18.85927168084700490843340412488, 19.88636005144530613870746315734
