| L(s) = 1 | − 19·2-s − 162·3-s + 429·4-s + 3.07e3·6-s + 1.18e4·7-s − 1.91e4·8-s + 1.96e4·9-s + 3.54e4·11-s − 6.94e4·12-s − 1.43e5·13-s − 2.25e5·14-s + 3.16e5·16-s − 3.85e5·17-s − 3.73e5·18-s − 4.03e5·19-s − 1.92e6·21-s − 6.74e5·22-s − 2.23e5·23-s + 3.10e6·24-s + 2.72e6·26-s − 2.12e6·27-s + 5.09e6·28-s − 7.45e4·29-s − 5.02e6·31-s − 6.50e6·32-s − 5.74e6·33-s + 7.31e6·34-s + ⋯ |
| L(s) = 1 | − 0.839·2-s − 1.15·3-s + 0.837·4-s + 0.969·6-s + 1.86·7-s − 1.65·8-s + 9-s + 0.730·11-s − 0.967·12-s − 1.39·13-s − 1.56·14-s + 1.20·16-s − 1.11·17-s − 0.839·18-s − 0.709·19-s − 2.15·21-s − 0.613·22-s − 0.166·23-s + 1.91·24-s + 1.17·26-s − 0.769·27-s + 1.56·28-s − 0.0195·29-s − 0.977·31-s − 1.09·32-s − 0.843·33-s + 0.939·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 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
−11.98248971273409588617552340247, −11.65219751052240772245654576341, −11.31076411308324809241937054834, −11.01887205017104809519184776521, −9.977751454908935964891837066439, −9.875390233448212903863590403839, −8.730541396357450960022706104696, −8.616111132050934695687104718153, −7.75553495451413298600404374125, −6.99051637491127615614604873264, −6.69145548571116019382817309832, −5.88069703652180196630828963952, −5.13173395003014349065917905672, −4.70012198013035439341221857504, −3.81008043728244939869250272631, −2.47758048731764178809166918022, −1.87181347393320650828495340716, −1.25774947916263775721049213561, 0, 0,
1.25774947916263775721049213561, 1.87181347393320650828495340716, 2.47758048731764178809166918022, 3.81008043728244939869250272631, 4.70012198013035439341221857504, 5.13173395003014349065917905672, 5.88069703652180196630828963952, 6.69145548571116019382817309832, 6.99051637491127615614604873264, 7.75553495451413298600404374125, 8.616111132050934695687104718153, 8.730541396357450960022706104696, 9.875390233448212903863590403839, 9.977751454908935964891837066439, 11.01887205017104809519184776521, 11.31076411308324809241937054834, 11.65219751052240772245654576341, 11.98248971273409588617552340247
