| L(s) = 1 | + 15·4-s − 9·9-s + 22·11-s + 161·16-s − 280·19-s − 300·29-s − 256·31-s − 135·36-s − 236·41-s + 330·44-s − 610·49-s + 1.48e3·59-s + 244·61-s + 1.45e3·64-s − 1.97e3·71-s − 4.20e3·76-s − 2.20e3·79-s + 81·81-s + 940·89-s − 198·99-s + 3.00e3·101-s + 4.38e3·109-s − 4.50e3·116-s + 363·121-s − 3.84e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 1/3·9-s + 0.603·11-s + 2.51·16-s − 3.38·19-s − 1.92·29-s − 1.48·31-s − 5/8·36-s − 0.898·41-s + 1.13·44-s − 1.77·49-s + 3.26·59-s + 0.512·61-s + 2.84·64-s − 3.30·71-s − 6.33·76-s − 3.13·79-s + 1/9·81-s + 1.11·89-s − 0.201·99-s + 2.95·101-s + 3.84·109-s − 3.60·116-s + 3/11·121-s − 2.78·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.225918101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.225918101\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 610 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4390 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5470 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19710 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 p^{2} T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 102670 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291030 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 740 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 586150 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 988 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 778030 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1100 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 390150 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 470 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 418750 T^{2} + p^{6} T^{4} \) |
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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
−10.27974751694400137312139795956, −9.933079419289534288369554854077, −8.918334055496485553458204346740, −8.697097780186471783881046135554, −8.580385133159884591494865182172, −7.58528124903231350113213555905, −7.57009251904567912956849743678, −6.94354992980385881650851553884, −6.61288404838827388917823716989, −6.14731655937689618778257540320, −5.90648351793450641598580544304, −5.38237693049968110083686631504, −4.63082293161128468797669799173, −3.96163909793507138156104933663, −3.63024217310991587771042707689, −2.99155539733255144283625123537, −2.25616368766992110717609679843, −1.90092904012246334966189776388, −1.60228214691283004127041661452, −0.34370911471092556443256149036,
0.34370911471092556443256149036, 1.60228214691283004127041661452, 1.90092904012246334966189776388, 2.25616368766992110717609679843, 2.99155539733255144283625123537, 3.63024217310991587771042707689, 3.96163909793507138156104933663, 4.63082293161128468797669799173, 5.38237693049968110083686631504, 5.90648351793450641598580544304, 6.14731655937689618778257540320, 6.61288404838827388917823716989, 6.94354992980385881650851553884, 7.57009251904567912956849743678, 7.58528124903231350113213555905, 8.580385133159884591494865182172, 8.697097780186471783881046135554, 8.918334055496485553458204346740, 9.933079419289534288369554854077, 10.27974751694400137312139795956
