| L(s) = 1 | − 3·2-s − 4·3-s + 8·4-s + 12·6-s − 45·8-s + 27·9-s − 32·12-s + 74·13-s + 135·16-s − 81·18-s + 180·24-s − 250·25-s − 222·26-s − 260·27-s − 282·29-s + 344·31-s − 360·32-s + 216·36-s − 296·39-s − 426·41-s − 48·47-s − 540·48-s − 686·49-s + 750·50-s + 592·52-s + 780·54-s + 846·58-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.769·3-s + 4-s + 0.816·6-s − 1.98·8-s + 9-s − 0.769·12-s + 1.57·13-s + 2.10·16-s − 1.06·18-s + 1.53·24-s − 2·25-s − 1.67·26-s − 1.85·27-s − 1.80·29-s + 1.99·31-s − 1.98·32-s + 36-s − 1.21·39-s − 1.62·41-s − 0.148·47-s − 1.62·48-s − 2·49-s + 2.12·50-s + 1.57·52-s + 1.96·54-s + 1.91·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 23 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 74 T + 3279 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 282 T + 55135 T^{2} + 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 344 T + 88545 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 426 T + 112555 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 48 T - 101519 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 1176 T + 1025065 T^{2} + 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 1226 T + 1114059 T^{2} - 1226 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−10.30792866267953517781133355327, −9.650167525094752719665546526069, −9.375203730262149961250469711864, −9.142727367362008080403396216415, −8.273854839377873230986918143317, −8.036041524584499579463020397091, −7.75673856470330812448552301857, −6.90344699600709196382596827894, −6.55672216130415090302545912052, −6.13306054101133600683790892643, −5.77099939202915742058881016114, −5.33942801291885513262722420341, −4.45490125756870609383157900573, −3.62451362876479624388955763697, −3.53834289405068283551636060737, −2.52707784573468122735832833954, −1.62477027138258993685381313017, −1.33146011199116449577164528096, 0, 0,
1.33146011199116449577164528096, 1.62477027138258993685381313017, 2.52707784573468122735832833954, 3.53834289405068283551636060737, 3.62451362876479624388955763697, 4.45490125756870609383157900573, 5.33942801291885513262722420341, 5.77099939202915742058881016114, 6.13306054101133600683790892643, 6.55672216130415090302545912052, 6.90344699600709196382596827894, 7.75673856470330812448552301857, 8.036041524584499579463020397091, 8.273854839377873230986918143317, 9.142727367362008080403396216415, 9.375203730262149961250469711864, 9.650167525094752719665546526069, 10.30792866267953517781133355327
