| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.728746279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.728746279\) |
| \(L(\frac{3}{2})\) |
|
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
−8.724492246229974815332020431449, −8.529988829823676052741559323287, −7.899976318518268196317260309242, −7.60462515660041835347924566117, −7.39908657434862823320392791395, −6.82904431082744527442095660219, −6.34540219751821015892409769748, −6.02745271972089421507969514298, −5.87484216211964272315935782236, −5.70345515407495021404163652622, −5.00586822143616922746744999482, −4.52279848803978774815142220243, −4.01674866907012371769941478268, −3.60641786610895876902494841490, −3.16511995193899276948389951815, −3.00606466374129565137092706707, −2.28914108661715918085780589450, −1.66409038449636696386426825227, −1.12427107808873043888609831643, −0.66549029873207549241152803058,
0.66549029873207549241152803058, 1.12427107808873043888609831643, 1.66409038449636696386426825227, 2.28914108661715918085780589450, 3.00606466374129565137092706707, 3.16511995193899276948389951815, 3.60641786610895876902494841490, 4.01674866907012371769941478268, 4.52279848803978774815142220243, 5.00586822143616922746744999482, 5.70345515407495021404163652622, 5.87484216211964272315935782236, 6.02745271972089421507969514298, 6.34540219751821015892409769748, 6.82904431082744527442095660219, 7.39908657434862823320392791395, 7.60462515660041835347924566117, 7.899976318518268196317260309242, 8.529988829823676052741559323287, 8.724492246229974815332020431449
