| L(s) = 1 | + 22·5-s + 359·25-s + 260·29-s − 460·41-s + 686·49-s − 1.66e3·61-s + 3.34e3·89-s − 1.19e3·101-s + 3.49e3·109-s − 2.66e3·121-s + 5.14e3·125-s + 127-s + 131-s + 137-s + 139-s + 5.72e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.07e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.96·5-s + 2.87·25-s + 1.66·29-s − 1.75·41-s + 2·49-s − 3.48·61-s + 3.97·89-s − 1.17·101-s + 3.06·109-s − 2·121-s + 3.68·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3.27·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.85·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.453870351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.453870351\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 22 T + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 94 T + p^{3} T^{2} )( 1 + 94 T + p^{3} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 130 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )( 1 + 214 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 230 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 518 T + p^{3} T^{2} )( 1 + 518 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1098 T + p^{3} T^{2} )( 1 + 1098 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1670 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )( 1 + 594 T + p^{3} T^{2} ) \) |
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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
−9.242606567388223089449599647044, −8.890496686440826494482025201192, −8.886388621346417316408307534218, −8.302487456275872774522789583781, −7.59535219715772785118389792195, −7.43918566237217857947698655768, −6.72653752534507400394146930512, −6.30165611412243143443785106388, −6.27980778682208315304315181533, −5.72393850744701288430332696177, −5.07288415754077701735273686599, −4.99673806143160264075215229592, −4.46881877869629376655316573814, −3.74614084849373086636683863282, −3.09343047945337872246443849535, −2.75461304999521101785378732319, −2.18163134816094451808498914948, −1.69151459256242394577532870725, −1.17842050602854067873942581935, −0.53515995628464418432660124075,
0.53515995628464418432660124075, 1.17842050602854067873942581935, 1.69151459256242394577532870725, 2.18163134816094451808498914948, 2.75461304999521101785378732319, 3.09343047945337872246443849535, 3.74614084849373086636683863282, 4.46881877869629376655316573814, 4.99673806143160264075215229592, 5.07288415754077701735273686599, 5.72393850744701288430332696177, 6.27980778682208315304315181533, 6.30165611412243143443785106388, 6.72653752534507400394146930512, 7.43918566237217857947698655768, 7.59535219715772785118389792195, 8.302487456275872774522789583781, 8.886388621346417316408307534218, 8.890496686440826494482025201192, 9.242606567388223089449599647044
