| L(s) = 1 | − 6·3-s + 16·5-s − 2·7-s + 27·9-s − 22·11-s − 76·13-s − 96·15-s − 26·17-s + 54·19-s + 12·21-s − 224·23-s + 74·25-s − 108·27-s + 222·29-s + 40·31-s + 132·33-s − 32·35-s − 48·37-s + 456·39-s − 494·41-s + 66·43-s + 432·45-s + 64·47-s − 650·49-s + 156·51-s − 84·53-s − 352·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.43·5-s − 0.107·7-s + 9-s − 0.603·11-s − 1.62·13-s − 1.65·15-s − 0.370·17-s + 0.652·19-s + 0.124·21-s − 2.03·23-s + 0.591·25-s − 0.769·27-s + 1.42·29-s + 0.231·31-s + 0.696·33-s − 0.154·35-s − 0.213·37-s + 1.87·39-s − 1.88·41-s + 0.234·43-s + 1.43·45-s + 0.198·47-s − 1.89·49-s + 0.428·51-s − 0.217·53-s − 0.862·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 16 T + 182 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 76 T + 5310 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 54 T + 11774 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 222 T + 43642 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T - 29250 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 48 T + 85910 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 64 T + 189662 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 84 T + 164350 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 1104 T + 736358 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 928 T + 626214 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 456 T + 488494 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 592 T + 341742 T^{2} + 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 230 T + 954126 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 348 T + 307798 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.11016588891183107888154344013, −10.00975496386647978677834835633, −9.442732485679175872997419683629, −9.254525088331411924577258089752, −8.321671860512676192372816728977, −7.924512373185252262401273636452, −7.46886609319762748482861277275, −6.84959523401718471156734771170, −6.35230872150857586439864271099, −6.16200721750146439473014518873, −5.35836803531288767170612636688, −5.34605527422431501609677021062, −4.56082661505276639723512582519, −4.32805515397844322074878777777, −3.10080724396107140755644242478, −2.68543577132564273521205376323, −1.79911492739856569575008455635, −1.49550935135242168299182012784, 0, 0,
1.49550935135242168299182012784, 1.79911492739856569575008455635, 2.68543577132564273521205376323, 3.10080724396107140755644242478, 4.32805515397844322074878777777, 4.56082661505276639723512582519, 5.34605527422431501609677021062, 5.35836803531288767170612636688, 6.16200721750146439473014518873, 6.35230872150857586439864271099, 6.84959523401718471156734771170, 7.46886609319762748482861277275, 7.924512373185252262401273636452, 8.321671860512676192372816728977, 9.254525088331411924577258089752, 9.442732485679175872997419683629, 10.00975496386647978677834835633, 10.11016588891183107888154344013
