| L(s) = 1 | − 4·2-s − 2·3-s + 8·4-s + 34·5-s + 8·6-s − 20·7-s − 32·8-s + 27·9-s − 136·10-s + 32·11-s − 16·12-s − 91·13-s + 80·14-s − 68·15-s + 128·16-s + 13·17-s − 108·18-s − 30·19-s + 272·20-s + 40·21-s − 128·22-s − 78·23-s + 64·24-s + 617·25-s + 364·26-s − 154·27-s − 160·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.384·3-s + 4-s + 3.04·5-s + 0.544·6-s − 1.07·7-s − 1.41·8-s + 9-s − 4.30·10-s + 0.877·11-s − 0.384·12-s − 1.94·13-s + 1.52·14-s − 1.17·15-s + 2·16-s + 0.185·17-s − 1.41·18-s − 0.362·19-s + 3.04·20-s + 0.415·21-s − 1.24·22-s − 0.707·23-s + 0.544·24-s + 4.93·25-s + 2.74·26-s − 1.09·27-s − 1.07·28-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
| \(L(2)\) |
\(\approx\) |
\(0.5557716438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5557716438\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 17 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 32 T - 307 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 13 T - 4744 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T - 5959 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 78 T - 6083 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 197 T + 14420 T^{2} + 197 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 74 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 227 T + 876 T^{2} - 227 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 165 T - 41696 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 156 T - 55171 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 864 T + 541117 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 862 T + 442281 T^{2} + 862 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 654 T + 69805 T^{2} + 654 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 215 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 628 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 266 T - 634213 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 238 T - 856029 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−19.48087388305308515481482611374, −18.96574936302645102981319399967, −18.09254921048204300184648378740, −17.90997346769399799405801989332, −17.18157593915280664305678696405, −16.92124724524974774626837997417, −16.31112134567384953240925394430, −14.86177173843490193203706774909, −14.48917747574907568308876279142, −13.26329777588852698611397520397, −12.86927017481398855429709189815, −11.98747629761118495658865541461, −10.38286862212767469402173284607, −9.887502046509861969066788993308, −9.370262842225942902263192995265, −9.320592364977432349566439380192, −7.23055185159361865807648753128, −6.23548438188844542950081286302, −5.68016097036963500634962429051, −2.13695132026179111184784576896,
2.13695132026179111184784576896, 5.68016097036963500634962429051, 6.23548438188844542950081286302, 7.23055185159361865807648753128, 9.320592364977432349566439380192, 9.370262842225942902263192995265, 9.887502046509861969066788993308, 10.38286862212767469402173284607, 11.98747629761118495658865541461, 12.86927017481398855429709189815, 13.26329777588852698611397520397, 14.48917747574907568308876279142, 14.86177173843490193203706774909, 16.31112134567384953240925394430, 16.92124724524974774626837997417, 17.18157593915280664305678696405, 17.90997346769399799405801989332, 18.09254921048204300184648378740, 18.96574936302645102981319399967, 19.48087388305308515481482611374
