| L(s) = 1 | + 19·2-s + 429·4-s + 1.18e4·7-s + 1.91e4·8-s − 3.54e4·11-s − 1.43e5·13-s + 2.25e5·14-s + 3.16e5·16-s + 3.85e5·17-s − 4.03e5·19-s − 6.74e5·22-s + 2.23e5·23-s − 2.72e6·26-s + 5.09e6·28-s + 7.45e4·29-s − 5.02e6·31-s + 6.50e6·32-s + 7.31e6·34-s − 5.37e6·37-s − 7.66e6·38-s − 1.42e7·41-s − 2.77e7·43-s − 1.52e7·44-s + 4.25e6·46-s + 9.59e7·47-s + 2.87e7·49-s − 6.16e7·52-s + ⋯ |
| L(s) = 1 | + 0.839·2-s + 0.837·4-s + 1.86·7-s + 1.65·8-s − 0.730·11-s − 1.39·13-s + 1.56·14-s + 1.20·16-s + 1.11·17-s − 0.709·19-s − 0.613·22-s + 0.166·23-s − 1.17·26-s + 1.56·28-s + 0.0195·29-s − 0.977·31-s + 1.09·32-s + 0.939·34-s − 0.471·37-s − 0.596·38-s − 0.785·41-s − 1.23·43-s − 0.612·44-s + 0.139·46-s + 2.86·47-s + 0.711·49-s − 1.16·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.509948691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.509948691\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 19 T - 17 p^{2} T^{2} - 19 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1696 p T + 327218 p^{3} T^{2} - 1696 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 35488 T + 526013014 T^{2} + 35488 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 11052 p T + 24105149134 T^{2} + 11052 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 385156 T + 268539949078 T^{2} - 385156 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 403296 T + 684929514838 T^{2} + 403296 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 223704 T - 1375273107794 T^{2} - 223704 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 74572 T + 28833430018078 T^{2} - 74572 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5027128 T + 52415931233342 T^{2} + 5027128 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5373628 T + 231061724951934 T^{2} + 5373628 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14211332 T + 443988635955862 T^{2} + 14211332 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 27748920 T + 1170232974699430 T^{2} + 27748920 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 95966440 T + 4320659216802910 T^{2} - 95966440 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 64305596 T + 6611083028543086 T^{2} + 64305596 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 187863136 T + 23071633420288438 T^{2} + 187863136 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 154080060 T + 23302683905802238 T^{2} - 154080060 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 33592376 T - 10819815556424362 T^{2} + 33592376 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 228270976 T + 45777616900481806 T^{2} - 228270976 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 33122316 T + 68371107952007926 T^{2} - 33122316 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 932406760 T + 453226630902929438 T^{2} + 932406760 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 207040152 T + 372783310330485238 T^{2} - 207040152 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2522676 p T + 610925899926766678 T^{2} + 2522676 p^{10} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 387134596 T - 734969029248610362 T^{2} + 387134596 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.13273510292471023562818767264, −10.53472697429444211062498978496, −10.07054311986075153218788592540, −9.616408939282575629044628052325, −8.650728110007209542898953173338, −8.261306065617549893854654271662, −7.75864672138173073312680953202, −7.29954942395982781936269700194, −7.17364594916820585878831480961, −6.23235683213134233586545329347, −5.32461757183912240518117820847, −5.32150714205889296428938011606, −4.66958338820583730352030391153, −4.39148124738874526125396584609, −3.60599763762996168133480229799, −2.84275175356950736461879073738, −2.21104699988800856367735432181, −1.72587338301419173192860277018, −1.37092736223913751605471797823, −0.42150633163475647269788967840,
0.42150633163475647269788967840, 1.37092736223913751605471797823, 1.72587338301419173192860277018, 2.21104699988800856367735432181, 2.84275175356950736461879073738, 3.60599763762996168133480229799, 4.39148124738874526125396584609, 4.66958338820583730352030391153, 5.32150714205889296428938011606, 5.32461757183912240518117820847, 6.23235683213134233586545329347, 7.17364594916820585878831480961, 7.29954942395982781936269700194, 7.75864672138173073312680953202, 8.261306065617549893854654271662, 8.650728110007209542898953173338, 9.616408939282575629044628052325, 10.07054311986075153218788592540, 10.53472697429444211062498978496, 11.13273510292471023562818767264
