| L(s) = 1 | − 19·2-s + 429·4-s + 1.25e3·5-s − 1.18e4·7-s − 1.91e4·8-s − 2.37e4·10-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 + 5.36e5·20-s + 6.74e5·22-s − 2.23e5·23-s + 1.17e6·25-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 − 1.48e7·35-s + 5.37e6·37-s + 7.66e6·38-s − 2.39e7·40-s − 1.42e7·41-s + ⋯ |
| L(s) = 1 | − 0.839·2-s + 0.837·4-s + 0.894·5-s − 1.86·7-s − 1.65·8-s − 0.751·10-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 + 0.749·20-s + 0.613·22-s − 0.166·23-s + 3/5·25-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 − 1.67·35-s + 0.471·37-s + 0.596·38-s − 1.48·40-s − 0.785·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 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
−13.22507792103989415095560813895, −13.00499652022815517299775094807, −12.65656296573071284450520735566, −11.60895642450622763200767859346, −11.02586450709934580027793378523, −10.45924729045976592466477428275, −9.731498615145750304575527374568, −9.412384114134637543804760994796, −8.754397057336905430155646893131, −8.248680857667860900787177011590, −7.00112311358171124400084320701, −6.32824473247187077892180262011, −6.28133513688588776814669887265, −5.38467240505791407035913495882, −3.86224273573759658271626688990, −3.02883950645134132259172977111, −2.46626957077382352071554309710, −1.45105038198193150584580212025, 0, 0,
1.45105038198193150584580212025, 2.46626957077382352071554309710, 3.02883950645134132259172977111, 3.86224273573759658271626688990, 5.38467240505791407035913495882, 6.28133513688588776814669887265, 6.32824473247187077892180262011, 7.00112311358171124400084320701, 8.248680857667860900787177011590, 8.754397057336905430155646893131, 9.412384114134637543804760994796, 9.731498615145750304575527374568, 10.45924729045976592466477428275, 11.02586450709934580027793378523, 11.60895642450622763200767859346, 12.65656296573071284450520735566, 13.00499652022815517299775094807, 13.22507792103989415095560813895
