| L(s) = 1 | − 24·2-s − 174·3-s − 584·4-s + 1.25e3·5-s + 4.17e3·6-s + 4.80e3·7-s + 2.95e4·8-s + 6.66e3·9-s − 3.00e4·10-s + 1.85e4·11-s + 1.01e5·12-s − 5.10e4·13-s − 1.15e5·14-s − 2.17e5·15-s + 576·16-s − 3.73e5·17-s − 1.60e5·18-s − 1.43e5·19-s − 7.30e5·20-s − 8.35e5·21-s − 4.45e5·22-s − 4.98e5·23-s − 5.14e6·24-s + 1.17e6·25-s + 1.22e6·26-s − 4.77e5·27-s − 2.80e6·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.24·3-s − 1.14·4-s + 0.894·5-s + 1.31·6-s + 0.755·7-s + 2.55·8-s + 0.338·9-s − 0.948·10-s + 0.382·11-s + 1.41·12-s − 0.496·13-s − 0.801·14-s − 1.10·15-s + 0.00219·16-s − 1.08·17-s − 0.359·18-s − 0.252·19-s − 1.02·20-s − 0.937·21-s − 0.405·22-s − 0.371·23-s − 3.16·24-s + 3/5·25-s + 0.526·26-s − 0.172·27-s − 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p^{3} T + 145 p^{3} T^{2} + 3 p^{12} T^{3} + p^{18} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 58 p T + 2623 p^{2} T^{2} + 58 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 18566 T + 4136090463 T^{2} - 18566 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3930 p T + 19172461323 T^{2} + 3930 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 373910 T + 270283008251 T^{2} + 373910 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 143276 T + 642750798250 T^{2} + 143276 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 498908 T - 1733603053870 T^{2} + 498908 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 399226 p T + 61819642147595 T^{2} + 399226 p^{10} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3953760 T + 18380996896542 T^{2} + 3953760 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3205412 T + 134679597433390 T^{2} + 3205412 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1058992 T - 202350146478094 T^{2} - 1058992 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15948180 T + 312060834724314 T^{2} - 15948180 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 65501290 T + 2591944660543287 T^{2} - 65501290 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 25114688 T + 6536128371514410 T^{2} + 25114688 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 116159208 T + 8473255943386694 T^{2} + 116159208 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 44688544 T + 21891928402164378 T^{2} + 44688544 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 118092496 T + 11543378662237830 T^{2} - 118092496 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 294165824 T + 66419314231297806 T^{2} + 294165824 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 57419332 T + 116103868936695574 T^{2} + 57419332 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 692852854 T + 353229306838520119 T^{2} + 692852854 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6514216 p T + 378894185867322102 T^{2} + 6514216 p^{10} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 779043704 T + 776620283810146850 T^{2} + 779043704 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2673039406 T + 3290052660205451443 T^{2} + 2673039406 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−14.13385977967800453904270733142, −13.64664731359557892116212425239, −12.90639181970161657491855120227, −12.55884841704641277353379114775, −11.31771657356272869470115669982, −11.09525134376827389948380142627, −10.37175410722683179387072686485, −9.731665429943266820692532590976, −8.964318341000569705781051570471, −8.905595636040902495495870369867, −7.68905770628825976643235786050, −7.19428124837091048835379462598, −5.66046106946367216334775453763, −5.65037960346970989466098281220, −4.59274530100514490928847016055, −4.02160144344251659376974537382, −1.97470375620916312201725157159, −1.30518867075844682515039124990, 0, 0,
1.30518867075844682515039124990, 1.97470375620916312201725157159, 4.02160144344251659376974537382, 4.59274530100514490928847016055, 5.65037960346970989466098281220, 5.66046106946367216334775453763, 7.19428124837091048835379462598, 7.68905770628825976643235786050, 8.905595636040902495495870369867, 8.964318341000569705781051570471, 9.731665429943266820692532590976, 10.37175410722683179387072686485, 11.09525134376827389948380142627, 11.31771657356272869470115669982, 12.55884841704641277353379114775, 12.90639181970161657491855120227, 13.64664731359557892116212425239, 14.13385977967800453904270733142
