L(s) = 1 | − 3-s + 4-s − 2·9-s + 5·11-s − 12-s − 3·16-s + 5·23-s + 5·27-s + 5·31-s − 5·33-s − 2·36-s + 9·37-s + 5·44-s + 5·47-s + 3·48-s − 11·49-s − 20·59-s − 7·64-s + 16·67-s − 5·69-s − 10·71-s + 81-s + 15·89-s + 5·92-s − 5·93-s − 9·97-s − 10·99-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/2·4-s − 2/3·9-s + 1.50·11-s − 0.288·12-s − 3/4·16-s + 1.04·23-s + 0.962·27-s + 0.898·31-s − 0.870·33-s − 1/3·36-s + 1.47·37-s + 0.753·44-s + 0.729·47-s + 0.433·48-s − 1.57·49-s − 2.60·59-s − 7/8·64-s + 1.95·67-s − 0.601·69-s − 1.18·71-s + 1/9·81-s + 1.58·89-s + 0.521·92-s − 0.518·93-s − 0.913·97-s − 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.791076603\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.791076603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.334116700602520655141106307490, −7.88189984639950325821541645080, −7.36363417350411796388354217724, −6.78418149828506902273767110526, −6.54198394299290963386083702175, −6.16413324110360088746870153079, −5.77283528046694900470934438747, −5.09364022756580265241522237295, −4.50472901188592223769843382490, −4.34327717232276938172334803376, −3.36175519863667810144112047102, −3.02646002813767196876123634257, −2.31580627630570792280911220605, −1.52592130713094820649462742500, −0.71797459166103602003118727466,
0.71797459166103602003118727466, 1.52592130713094820649462742500, 2.31580627630570792280911220605, 3.02646002813767196876123634257, 3.36175519863667810144112047102, 4.34327717232276938172334803376, 4.50472901188592223769843382490, 5.09364022756580265241522237295, 5.77283528046694900470934438747, 6.16413324110360088746870153079, 6.54198394299290963386083702175, 6.78418149828506902273767110526, 7.36363417350411796388354217724, 7.88189984639950325821541645080, 8.334116700602520655141106307490