L(s) = 1 | + 4-s + 4·9-s − 2·11-s + 16-s + 8·23-s + 2·25-s + 4·36-s + 20·37-s − 2·44-s + 4·53-s + 64-s − 24·67-s + 24·71-s + 7·81-s + 8·92-s − 8·99-s + 2·100-s − 24·113-s − 7·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 20·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 4/3·9-s − 0.603·11-s + 1/4·16-s + 1.66·23-s + 2/5·25-s + 2/3·36-s + 3.28·37-s − 0.301·44-s + 0.549·53-s + 1/8·64-s − 2.93·67-s + 2.84·71-s + 7/9·81-s + 0.834·92-s − 0.804·99-s + 1/5·100-s − 2.25·113-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 1.64·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.026507316\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.026507316\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
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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
−7.918950853808782972780477870138, −7.59352079059188887379684491445, −7.14278878197371069934491200886, −6.86087138996690322550355374245, −6.30082857681757246165778000841, −5.96807145758573138633356947474, −5.28993606316965103220242722774, −4.90474793617914491960614423064, −4.38838165650365730350397869180, −4.02504158971345539207936180833, −3.26143044986238475310620392100, −2.73466801355139759125384496825, −2.30824220791518763375460369669, −1.39259412968020420261479324692, −0.867271341589148695159958215278,
0.867271341589148695159958215278, 1.39259412968020420261479324692, 2.30824220791518763375460369669, 2.73466801355139759125384496825, 3.26143044986238475310620392100, 4.02504158971345539207936180833, 4.38838165650365730350397869180, 4.90474793617914491960614423064, 5.28993606316965103220242722774, 5.96807145758573138633356947474, 6.30082857681757246165778000841, 6.86087138996690322550355374245, 7.14278878197371069934491200886, 7.59352079059188887379684491445, 7.918950853808782972780477870138