| L(s) = 1 | + 2·2-s − 2·4-s − 8·8-s + 6·9-s − 32·11-s − 4·16-s − 8·17-s + 12·18-s + 32·19-s − 64·22-s + 28·25-s − 8·32-s − 16·34-s − 12·36-s + 64·38-s + 40·41-s + 32·43-s + 64·44-s + 76·49-s + 56·50-s − 128·59-s − 24·64-s − 256·67-s + 16·68-s − 48·72-s + 200·73-s − 64·76-s + ⋯ |
| L(s) = 1 | + 2-s − 1/2·4-s − 8-s + 2/3·9-s − 2.90·11-s − 1/4·16-s − 0.470·17-s + 2/3·18-s + 1.68·19-s − 2.90·22-s + 1.11·25-s − 1/4·32-s − 0.470·34-s − 1/3·36-s + 1.68·38-s + 0.975·41-s + 0.744·43-s + 1.45·44-s + 1.55·49-s + 1.11·50-s − 2.16·59-s − 3/8·64-s − 3.82·67-s + 4/17·68-s − 2/3·72-s + 2.73·73-s − 0.842·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8835630003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8835630003\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( 1 - p T + 3 p T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 390 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 64 T + 7554 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 80 T + 14610 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.38007144419951117460720073910, −13.01856626614660602989844745783, −12.63206147962328451585534254910, −12.40726944006768489398401158527, −12.09426334571972855301389625462, −11.74012896545958677224446927740, −10.80626742570984102236577363067, −10.76293102666116434721923976255, −10.70464897985039189470434307052, −10.19749434703203209978610563842, −9.496348007137509508991465712168, −9.284699578591150270908648951262, −9.110724250132546308985061125581, −8.293193361109677564047267773112, −7.82621438104503750076123410916, −7.77757615597159208351556357411, −7.16093454727449572360219821350, −6.74789168780566424944579060693, −5.78138868873153761470703667285, −5.55622303442588941650820156498, −5.14059384744607714226052879422, −4.61158122069529405690835623875, −4.24261628617026933892896956709, −3.19126847826640312997642897644, −2.68155907806404530091965479096,
2.68155907806404530091965479096, 3.19126847826640312997642897644, 4.24261628617026933892896956709, 4.61158122069529405690835623875, 5.14059384744607714226052879422, 5.55622303442588941650820156498, 5.78138868873153761470703667285, 6.74789168780566424944579060693, 7.16093454727449572360219821350, 7.77757615597159208351556357411, 7.82621438104503750076123410916, 8.293193361109677564047267773112, 9.110724250132546308985061125581, 9.284699578591150270908648951262, 9.496348007137509508991465712168, 10.19749434703203209978610563842, 10.70464897985039189470434307052, 10.76293102666116434721923976255, 10.80626742570984102236577363067, 11.74012896545958677224446927740, 12.09426334571972855301389625462, 12.40726944006768489398401158527, 12.63206147962328451585534254910, 13.01856626614660602989844745783, 13.38007144419951117460720073910
