| L(s) = 1 | − 2-s + 3·5-s − 7-s + 8-s − 3·10-s − 3·11-s − 2·13-s + 14-s − 16-s + 3·17-s + 7·19-s + 3·22-s − 9·23-s + 5·25-s + 2·26-s − 6·29-s − 8·31-s − 3·34-s − 3·35-s + 37-s − 7·38-s + 3·40-s − 6·41-s − 2·43-s + 9·46-s − 6·49-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.904·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s − 1.87·23-s + 25-s + 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.514·34-s − 0.507·35-s + 0.164·37-s − 1.13·38-s + 0.474·40-s − 0.937·41-s − 0.304·43-s + 1.32·46-s − 6/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.08193305818444774546168739273, −9.556983611093547240577746528942, −9.455967414600466782049884595482, −8.858715184913233297577231728170, −8.541776383106467103341770360383, −7.80248977176881529367216742748, −7.62872203459024012761075058413, −7.33918464000050529623638159710, −6.77387675113480571165664383009, −6.03555339202875936953791011893, −5.85774895963751147474830638234, −5.48282874177802778030289571383, −4.96656064777934098147169083377, −4.58475787561205409920303095381, −3.50744770951295960094005131160, −3.44579368893884223304965739463, −2.61916154080112620792118898406, −1.90125293647238192678882950015, −1.68832937132653323321807153076, −0.50027483335206789807484504301,
0.50027483335206789807484504301, 1.68832937132653323321807153076, 1.90125293647238192678882950015, 2.61916154080112620792118898406, 3.44579368893884223304965739463, 3.50744770951295960094005131160, 4.58475787561205409920303095381, 4.96656064777934098147169083377, 5.48282874177802778030289571383, 5.85774895963751147474830638234, 6.03555339202875936953791011893, 6.77387675113480571165664383009, 7.33918464000050529623638159710, 7.62872203459024012761075058413, 7.80248977176881529367216742748, 8.541776383106467103341770360383, 8.858715184913233297577231728170, 9.455967414600466782049884595482, 9.556983611093547240577746528942, 10.08193305818444774546168739273
