| L(s) = 1 | + 2-s − 4-s − 3·8-s − 6·9-s − 16-s − 6·18-s − 4·29-s + 5·32-s + 6·36-s − 12·37-s − 7·49-s + 20·53-s − 4·58-s + 7·64-s + 18·72-s − 12·74-s + 27·81-s − 7·98-s + 20·106-s − 36·109-s − 4·113-s + 4·116-s − 6·121-s + 127-s − 3·128-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2·9-s − 1/4·16-s − 1.41·18-s − 0.742·29-s + 0.883·32-s + 36-s − 1.97·37-s − 49-s + 2.74·53-s − 0.525·58-s + 7/8·64-s + 2.12·72-s − 1.39·74-s + 3·81-s − 0.707·98-s + 1.94·106-s − 3.44·109-s − 0.376·113-s + 0.371·116-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9889009200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9889009200\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−10.63468448071673577933936047288, −10.39490492758175891975624130811, −9.720647912280301973090157850757, −9.117044024240259598992490925138, −9.100642082825514916572172952544, −8.451620285226082817438238383828, −8.264759705607414418084218435285, −7.74945170469985958975756886581, −6.97339920995615534219117433044, −6.65588394753073699105763174416, −5.96423572721207226043951263005, −5.67559950198381488768929830048, −5.16970240960037055880927422595, −5.02463829004666174347507188398, −3.93794448326983746645759521564, −3.84945677963745748844522413355, −3.00786857940741681372377139722, −2.74011743395170346053546973987, −1.84703248994220899101938550711, −0.45354039279428493051328016724,
0.45354039279428493051328016724, 1.84703248994220899101938550711, 2.74011743395170346053546973987, 3.00786857940741681372377139722, 3.84945677963745748844522413355, 3.93794448326983746645759521564, 5.02463829004666174347507188398, 5.16970240960037055880927422595, 5.67559950198381488768929830048, 5.96423572721207226043951263005, 6.65588394753073699105763174416, 6.97339920995615534219117433044, 7.74945170469985958975756886581, 8.264759705607414418084218435285, 8.451620285226082817438238383828, 9.100642082825514916572172952544, 9.117044024240259598992490925138, 9.720647912280301973090157850757, 10.39490492758175891975624130811, 10.63468448071673577933936047288
