| L(s) = 1 | + 15·2-s + 40·3-s + 75·4-s + 600·6-s − 600·7-s + 795·8-s − 2.52e3·9-s + 4.34e3·11-s + 3.00e3·12-s + 1.76e4·13-s − 9.00e3·14-s + 1.30e4·16-s + 6.87e3·17-s − 3.78e4·18-s + 1.82e4·19-s − 2.40e4·21-s + 6.51e4·22-s + 2.11e4·23-s + 3.18e4·24-s + 2.65e5·26-s − 1.78e5·27-s − 4.50e4·28-s + 5.58e4·29-s − 3.01e5·31-s − 2.46e4·32-s + 1.73e5·33-s + 1.03e5·34-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.855·3-s + 0.585·4-s + 1.13·6-s − 0.661·7-s + 0.548·8-s − 1.15·9-s + 0.984·11-s + 0.501·12-s + 2.23·13-s − 0.876·14-s + 0.799·16-s + 0.339·17-s − 1.53·18-s + 0.608·19-s − 0.565·21-s + 1.30·22-s + 0.361·23-s + 0.469·24-s + 2.95·26-s − 1.74·27-s − 0.387·28-s + 0.424·29-s − 1.81·31-s − 0.132·32-s + 0.841·33-s + 0.449·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.915926421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.915926421\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 15 T + 75 p T^{2} - 15 p^{7} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 40 T + 1375 p T^{2} - 40 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 600 T + 592250 T^{2} + 600 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4344 T + 33551301 T^{2} - 4344 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1360 p T + 176633850 T^{2} - 1360 p^{8} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6870 T + 752062875 T^{2} - 6870 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 18200 T + 1670645253 T^{2} - 18200 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 21120 T - 431976950 T^{2} - 21120 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 55800 T + 7178799018 T^{2} - 55800 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 301776 T + 74506854266 T^{2} + 301776 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 609860 T + 262289383950 T^{2} - 609860 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2646 p T + 391801850811 T^{2} + 2646 p^{8} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 966400 T + 711647085750 T^{2} - 966400 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 38040 p T + 1782082095150 T^{2} - 38040 p^{8} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 130740 T + 2038893798750 T^{2} - 130740 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2067600 T + 4803250907238 T^{2} - 2067600 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 582044 T - 1843726773474 T^{2} - 582044 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 255720 T + 11262543795125 T^{2} - 255720 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4728216 T + 23373621952446 T^{2} + 4728216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1339430 T + 21386673483675 T^{2} + 1339430 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7186200 T + 50054286054218 T^{2} + 7186200 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12049560 T + 89789116985725 T^{2} - 12049560 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5990850 T + 65725956363283 T^{2} + 5990850 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17120020 T + 232426185762150 T^{2} + 17120020 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−16.31845651099133721360820333849, −15.51518151933933341231219317522, −14.55218128236128044839113501202, −14.45041124524007586912210027017, −13.77942353725980860137404566346, −13.35562242036575649682662563033, −12.86144526156551222849648386674, −11.93888218892642804884787299832, −11.20308232999855585134129311290, −10.70974293031736466494808325503, −9.281635666233924401419681334715, −9.009660049507735027659767635658, −8.150319843154841174117151732876, −7.15603504233198576598314518168, −5.89989530664947875835339378986, −5.70731033312829797868603045237, −4.04318090648680426860868469488, −3.67412521933221315571864002149, −2.73539118290749789693889009186, −1.11290665408138109805161856361,
1.11290665408138109805161856361, 2.73539118290749789693889009186, 3.67412521933221315571864002149, 4.04318090648680426860868469488, 5.70731033312829797868603045237, 5.89989530664947875835339378986, 7.15603504233198576598314518168, 8.150319843154841174117151732876, 9.009660049507735027659767635658, 9.281635666233924401419681334715, 10.70974293031736466494808325503, 11.20308232999855585134129311290, 11.93888218892642804884787299832, 12.86144526156551222849648386674, 13.35562242036575649682662563033, 13.77942353725980860137404566346, 14.45041124524007586912210027017, 14.55218128236128044839113501202, 15.51518151933933341231219317522, 16.31845651099133721360820333849
