| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s + 8·5-s − 24·6-s + 12·7-s + 32·8-s + 27·9-s + 32·10-s + 48·11-s − 72·12-s + 84·13-s + 48·14-s − 48·15-s + 80·16-s + 104·17-s + 108·18-s + 28·19-s + 96·20-s − 72·21-s + 192·22-s + 46·23-s − 192·24-s − 74·25-s + 336·26-s − 108·27-s + 144·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.715·5-s − 1.63·6-s + 0.647·7-s + 1.41·8-s + 9-s + 1.01·10-s + 1.31·11-s − 1.73·12-s + 1.79·13-s + 0.916·14-s − 0.826·15-s + 5/4·16-s + 1.48·17-s + 1.41·18-s + 0.338·19-s + 1.07·20-s − 0.748·21-s + 1.86·22-s + 0.417·23-s − 1.63·24-s − 0.591·25-s + 2.53·26-s − 0.769·27-s + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.921433364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.921433364\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 12 T + 594 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 48 T + 1190 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 84 T + 4110 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 104 T + 11378 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 28 T + 12762 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 132 T + 40334 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 46798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 324 T + 65598 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 140 T + 124310 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 548 T + 218602 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 464 T + 256862 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 336 T + 110810 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 72 T - 79978 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 60 T + 184014 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 444 T + 557498 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 672 T + 427310 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 252 T + 447798 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1692 T + 1701666 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 400 T + 192342 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 336 T + 1391954 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 220 T + 976774 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.06297512713794133088165406303, −12.39478350542303761457797894326, −11.77132133178420240247562408527, −11.74913162801284593794002580579, −11.18440993382823306071536531247, −10.68506598721816554621254007694, −9.938604545736183278032629847190, −9.776743138095994712330981548737, −8.546084886113601534665212586362, −8.281076446293099021037767332728, −7.14585402216190036455212162180, −6.80208346104643020606869581440, −5.99467261876733706251465533603, −5.94226890369352567012204560882, −5.12393301609889072256066311336, −4.68803834745659696567350859198, −3.68932868994594777497651311234, −3.33485659241343596257120943457, −1.55272768742799469341643332450, −1.37973425137806912957340102785,
1.37973425137806912957340102785, 1.55272768742799469341643332450, 3.33485659241343596257120943457, 3.68932868994594777497651311234, 4.68803834745659696567350859198, 5.12393301609889072256066311336, 5.94226890369352567012204560882, 5.99467261876733706251465533603, 6.80208346104643020606869581440, 7.14585402216190036455212162180, 8.281076446293099021037767332728, 8.546084886113601534665212586362, 9.776743138095994712330981548737, 9.938604545736183278032629847190, 10.68506598721816554621254007694, 11.18440993382823306071536531247, 11.74913162801284593794002580579, 11.77132133178420240247562408527, 12.39478350542303761457797894326, 13.06297512713794133088165406303
