| L(s) = 1 | − 2·5-s + 4·7-s + 2·9-s − 2·11-s − 8·13-s + 8·17-s + 3·25-s + 4·29-s − 8·35-s − 4·37-s + 12·41-s + 12·43-s − 4·45-s − 2·49-s + 12·53-s + 4·55-s + 8·59-s + 4·61-s + 8·63-s + 16·65-s − 8·67-s − 8·73-s − 8·77-s − 8·79-s − 5·81-s + 12·83-s − 16·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 2/3·9-s − 0.603·11-s − 2.21·13-s + 1.94·17-s + 3/5·25-s + 0.742·29-s − 1.35·35-s − 0.657·37-s + 1.87·41-s + 1.82·43-s − 0.596·45-s − 2/7·49-s + 1.64·53-s + 0.539·55-s + 1.04·59-s + 0.512·61-s + 1.00·63-s + 1.98·65-s − 0.977·67-s − 0.936·73-s − 0.911·77-s − 0.900·79-s − 5/9·81-s + 1.31·83-s − 1.73·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939577378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939577378\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.29337745529042836686368322725, −9.979827592003459750366128401511, −9.647300007049299126630998049989, −9.110914037463569081913808669818, −8.439327731050135934106151039587, −8.215925618402231751084931160980, −7.60905599352842779529658382068, −7.45947244689422514969178847370, −7.33350210153434649133865374924, −6.68526246485527454574842777213, −5.64585821283507488847693201437, −5.59308881404492628698672472052, −4.93436284945635794475446698267, −4.56549313646993592510363641777, −4.24678817438831924093805245995, −3.55755998642489480675553342667, −2.74486481252625911908793365479, −2.42932451957451320618097003376, −1.49815823732743381030610816745, −0.71846536667799770505343771019,
0.71846536667799770505343771019, 1.49815823732743381030610816745, 2.42932451957451320618097003376, 2.74486481252625911908793365479, 3.55755998642489480675553342667, 4.24678817438831924093805245995, 4.56549313646993592510363641777, 4.93436284945635794475446698267, 5.59308881404492628698672472052, 5.64585821283507488847693201437, 6.68526246485527454574842777213, 7.33350210153434649133865374924, 7.45947244689422514969178847370, 7.60905599352842779529658382068, 8.215925618402231751084931160980, 8.439327731050135934106151039587, 9.110914037463569081913808669818, 9.647300007049299126630998049989, 9.979827592003459750366128401511, 10.29337745529042836686368322725
