| L(s) = 1 | − 4·2-s + 8·4-s − 58·7-s + 236·11-s − 138·13-s + 232·14-s − 64·16-s − 542·17-s − 944·22-s + 538·23-s + 552·26-s − 464·28-s + 404·31-s + 256·32-s + 2.16e3·34-s + 1.30e3·37-s − 3.36e3·41-s − 2.17e3·43-s + 1.88e3·44-s − 2.15e3·46-s + 2.53e3·47-s + 1.68e3·49-s − 1.10e3·52-s − 1.22e3·53-s − 1.11e4·61-s − 1.61e3·62-s − 512·64-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 1.18·7-s + 1.95·11-s − 0.816·13-s + 1.18·14-s − 1/4·16-s − 1.87·17-s − 1.95·22-s + 1.01·23-s + 0.816·26-s − 0.591·28-s + 0.420·31-s + 1/4·32-s + 1.87·34-s + 0.951·37-s − 2.00·41-s − 1.17·43-s + 0.975·44-s − 1.01·46-s + 1.14·47-s + 0.700·49-s − 0.408·52-s − 0.435·53-s − 3.00·61-s − 0.420·62-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1168128517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1168128517\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 58 T + 1682 T^{2} + 58 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 118 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 138 T + 9522 T^{2} + 138 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 542 T + 146882 T^{2} + 542 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 182242 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 538 T + 144722 T^{2} - 538 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 952162 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 202 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 1302 T + 847602 T^{2} - 1302 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1682 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2178 T + 2371842 T^{2} + 2178 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 p T + 1458 p^{2} T^{2} - 54 p^{5} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1222 T + 746642 T^{2} + 1222 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22889122 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5598 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1502 T + 1128002 T^{2} - 1502 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6442 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5902 T + 17416802 T^{2} - 5902 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33613438 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12462 T + 77650722 T^{2} + 12462 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 84185918 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14622 T + 106901442 T^{2} - 14622 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−10.68026061022661309246544134996, −10.05745824261004230157177108356, −9.729355636127897649246074606356, −9.211733179136069002121473225882, −9.067165763001055126215035977807, −8.682367409212435790514922580595, −8.127083780804767862974547015554, −7.34262381804799873765635483034, −7.01783463116910347528451890623, −6.56449157097119528524537940846, −6.38456317163692058870175030304, −5.71161739128059005564576660639, −4.70062513132449074238973268686, −4.50950292347710796892003045348, −3.76536177193546086070567499563, −3.10985372230527298396724436900, −2.55297864894556280444109953981, −1.68467379930138418247203859424, −1.16105588966488958368139802314, −0.11856132610668302561315733702,
0.11856132610668302561315733702, 1.16105588966488958368139802314, 1.68467379930138418247203859424, 2.55297864894556280444109953981, 3.10985372230527298396724436900, 3.76536177193546086070567499563, 4.50950292347710796892003045348, 4.70062513132449074238973268686, 5.71161739128059005564576660639, 6.38456317163692058870175030304, 6.56449157097119528524537940846, 7.01783463116910347528451890623, 7.34262381804799873765635483034, 8.127083780804767862974547015554, 8.682367409212435790514922580595, 9.067165763001055126215035977807, 9.211733179136069002121473225882, 9.729355636127897649246074606356, 10.05745824261004230157177108356, 10.68026061022661309246544134996
