| L(s) = 1 | + 4·2-s + 12·4-s − 26·5-s + 14·7-s + 32·8-s − 104·10-s − 22·11-s + 2·13-s + 56·14-s + 80·16-s + 88·17-s + 42·19-s − 312·20-s − 88·22-s − 160·23-s + 294·25-s + 8·26-s + 168·28-s − 24·29-s + 248·31-s + 192·32-s + 352·34-s − 364·35-s − 52·37-s + 168·38-s − 832·40-s − 528·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 2.32·5-s + 0.755·7-s + 1.41·8-s − 3.28·10-s − 0.603·11-s + 0.0426·13-s + 1.06·14-s + 5/4·16-s + 1.25·17-s + 0.507·19-s − 3.48·20-s − 0.852·22-s − 1.45·23-s + 2.35·25-s + 0.0603·26-s + 1.13·28-s − 0.153·29-s + 1.43·31-s + 1.06·32-s + 1.77·34-s − 1.75·35-s − 0.231·37-s + 0.717·38-s − 3.28·40-s − 2.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 26 T + 382 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 4358 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 88 T + 11170 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 42 T + 802 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 160 T + 30142 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 24 T + 47590 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 p T + 67706 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 52 T + 72974 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 528 T + 186226 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 204 T + 162166 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 24 T - 91910 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 248 T + 234838 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 570 T + 465010 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1338 T + 881950 T^{2} + 1338 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 180 T + 566854 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 60 T - 115778 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 770786 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1928 T + 1782174 T^{2} + 1928 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 542 T + 439090 T^{2} - 542 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 140 T - 185930 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2712 T + 3415294 T^{2} + 2712 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.541452595026623145753714717940, −8.468067143921974384547627892327, −7.932508829555480529214763477861, −7.84328956147986122418428628559, −7.32806771349578582621129145263, −7.21109616878248172220563272461, −6.35214729873755987110283393610, −6.15548459718908900207225336602, −5.45316203068917228116597416489, −5.11789743783986242838198269569, −4.60729739367668293744682424140, −4.39156707743990234438521702720, −3.79263319283344767742745273745, −3.55451545711690551346189638589, −2.98633045924799306051447879768, −2.67648630289064019314556139999, −1.54076618430407156915559271304, −1.35471674237487864265329042804, 0, 0,
1.35471674237487864265329042804, 1.54076618430407156915559271304, 2.67648630289064019314556139999, 2.98633045924799306051447879768, 3.55451545711690551346189638589, 3.79263319283344767742745273745, 4.39156707743990234438521702720, 4.60729739367668293744682424140, 5.11789743783986242838198269569, 5.45316203068917228116597416489, 6.15548459718908900207225336602, 6.35214729873755987110283393610, 7.21109616878248172220563272461, 7.32806771349578582621129145263, 7.84328956147986122418428628559, 7.932508829555480529214763477861, 8.468067143921974384547627892327, 8.541452595026623145753714717940
