| L(s) = 1 | − 9·2-s − 375·4-s − 1.17e3·5-s − 4.80e3·7-s + 2.87e3·8-s + 1.05e4·10-s + 1.45e5·11-s + 8.65e4·13-s + 4.32e4·14-s − 7.55e4·16-s + 2.29e5·17-s − 2.21e5·19-s + 4.38e5·20-s − 1.31e6·22-s + 2.03e6·23-s + 2.85e3·25-s − 7.78e5·26-s + 1.80e6·28-s − 9.75e6·29-s + 2.04e5·31-s + 3.45e6·32-s − 2.06e6·34-s + 5.61e6·35-s − 1.39e7·37-s + 1.99e6·38-s − 3.35e6·40-s + 4.23e7·41-s + ⋯ |
| L(s) = 1 | − 0.397·2-s − 0.732·4-s − 0.837·5-s − 0.755·7-s + 0.247·8-s + 0.332·10-s + 3.00·11-s + 0.840·13-s + 0.300·14-s − 0.288·16-s + 0.667·17-s − 0.389·19-s + 0.613·20-s − 1.19·22-s + 1.51·23-s + 0.00145·25-s − 0.334·26-s + 0.553·28-s − 2.56·29-s + 0.0396·31-s + 0.582·32-s − 0.265·34-s + 0.632·35-s − 1.22·37-s + 0.154·38-s − 0.207·40-s + 2.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.057832870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057832870\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 9 T + 57 p^{3} T^{2} + 9 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 234 p T + 54642 p^{2} T^{2} + 234 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 145746 T + 9848423886 T^{2} - 145746 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 512 p^{2} T + 1774916238 p T^{2} - 512 p^{11} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 229842 T - 51722753798 T^{2} - 229842 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 221224 T + 34608506226 p T^{2} + 221224 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2035782 T + 4618931326270 T^{2} - 2035782 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9756252 T + 52470669069246 T^{2} + 9756252 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 204000 T + 30790423515710 T^{2} - 204000 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13959816 T + 306736999959926 T^{2} + 13959816 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 42362550 T + 1063812640967922 T^{2} - 42362550 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4763912 T + 345063686509734 T^{2} + 4763912 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 48278484 T + 2529195823681630 T^{2} - 48278484 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 108980352 T + 6796561315809670 T^{2} - 108980352 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 188376804 T + 23346971643078934 T^{2} - 188376804 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19722092 T + 14767573710118686 T^{2} - 19722092 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 70274396 T - 15240809102766810 T^{2} - 70274396 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 382044186 T + 120425517984233086 T^{2} - 382044186 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 191785896 T + 124167221244272702 T^{2} - 191785896 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 72592148 T + 239185581484447902 T^{2} + 72592148 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 187994232 T + 258704825037511030 T^{2} + 187994232 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 42930954 T + 403096805893580370 T^{2} + 42930954 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1726854096 T + 1929285565356657566 T^{2} - 1726854096 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−13.16262353250482083235715165076, −12.86829353606016304916117865145, −12.11576792728829831309338131489, −11.55842991260023374355475737868, −11.27017867916100748483309783450, −10.48679902028117155469819829460, −9.470437393377104650633155623789, −9.231868468424093347995746935994, −8.935263335945153620427384858850, −8.285488573675921924656200886445, −7.17485703241964063994260692264, −6.96030504125037825571951681638, −6.12621483782114098749204511629, −5.43151432532997204461146570863, −4.13569771824594429359871527472, −3.89954266606847188690879514360, −3.47547383741547718941506163587, −2.03021619482439936181469578411, −0.842743139598152433749507130465, −0.73025555212998984636711855164,
0.73025555212998984636711855164, 0.842743139598152433749507130465, 2.03021619482439936181469578411, 3.47547383741547718941506163587, 3.89954266606847188690879514360, 4.13569771824594429359871527472, 5.43151432532997204461146570863, 6.12621483782114098749204511629, 6.96030504125037825571951681638, 7.17485703241964063994260692264, 8.285488573675921924656200886445, 8.935263335945153620427384858850, 9.231868468424093347995746935994, 9.470437393377104650633155623789, 10.48679902028117155469819829460, 11.27017867916100748483309783450, 11.55842991260023374355475737868, 12.11576792728829831309338131489, 12.86829353606016304916117865145, 13.16262353250482083235715165076
