| L(s) = 1 | − 12·5-s + 24·7-s − 22·11-s − 8·13-s + 28·17-s + 68·19-s − 268·23-s − 18·25-s + 260·29-s − 112·31-s − 288·35-s + 316·37-s + 124·41-s + 148·43-s − 572·47-s − 130·49-s + 76·53-s + 264·55-s − 864·59-s − 136·61-s + 96·65-s + 512·67-s − 724·71-s + 972·73-s − 528·77-s + 528·79-s − 2.11e3·83-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 1.29·7-s − 0.603·11-s − 0.170·13-s + 0.399·17-s + 0.821·19-s − 2.42·23-s − 0.143·25-s + 1.66·29-s − 0.648·31-s − 1.39·35-s + 1.40·37-s + 0.472·41-s + 0.524·43-s − 1.77·47-s − 0.379·49-s + 0.196·53-s + 0.647·55-s − 1.90·59-s − 0.285·61-s + 0.183·65-s + 0.933·67-s − 1.21·71-s + 1.55·73-s − 0.781·77-s + 0.751·79-s − 2.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 24 T + 706 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 1310 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 28 T + 5558 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 68 T + 14378 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 268 T + 36214 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 260 T + 63694 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 112 T + 38414 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 316 T + 66254 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 124 T + 133750 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 74 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 572 T + 274438 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 76 T + 48098 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 864 T + 592918 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 136 T + 136062 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 512 T + 468662 T^{2} - 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 724 T + 694966 T^{2} + 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 972 T + 996374 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 528 T + 1052674 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2112 T + 2198694 T^{2} + 2112 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 288 T - 124782 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 500 T + 1245030 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.600178410141539583923507672087, −8.278595329809874528341132007612, −7.985852831573535686282338314890, −7.78244830374909943685011737400, −7.53580027999244594433788440635, −7.00575217938689067780002808988, −6.26277538124417949784904758899, −6.08243112500102668614039918615, −5.56324326272361904003357424859, −4.94827193607931544864424037013, −4.65395880685323897777534853217, −4.38572712428870159174072262960, −3.57516889948019528588592057699, −3.56510483458484869027119774808, −2.55890113389633905797180988300, −2.36095512734202074861320839503, −1.42891650320213566343461689452, −1.17760953987608182624534403609, 0, 0,
1.17760953987608182624534403609, 1.42891650320213566343461689452, 2.36095512734202074861320839503, 2.55890113389633905797180988300, 3.56510483458484869027119774808, 3.57516889948019528588592057699, 4.38572712428870159174072262960, 4.65395880685323897777534853217, 4.94827193607931544864424037013, 5.56324326272361904003357424859, 6.08243112500102668614039918615, 6.26277538124417949784904758899, 7.00575217938689067780002808988, 7.53580027999244594433788440635, 7.78244830374909943685011737400, 7.985852831573535686282338314890, 8.278595329809874528341132007612, 8.600178410141539583923507672087
