| L(s) = 1 | − 40·11-s + 168·19-s + 12·29-s − 448·31-s − 532·41-s − 610·49-s − 56·59-s + 364·61-s − 816·71-s − 96·79-s − 3.05e3·89-s − 2.49e3·101-s − 1.80e3·109-s − 1.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.65e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.09·11-s + 2.02·19-s + 0.0768·29-s − 2.59·31-s − 2.02·41-s − 1.77·49-s − 0.123·59-s + 0.764·61-s − 1.36·71-s − 0.136·79-s − 3.63·89-s − 2.45·101-s − 1.58·109-s − 1.09·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.754·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 610 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 20610 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1514050 T^{2} + p^{6} T^{4} \) |
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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
−9.458873460454900163766899739430, −9.270608277120423900144699495153, −8.556180195376830616700194636366, −8.328959020043083457743168771166, −7.62539202873709958105730203929, −7.58623751188550053148364349374, −6.92346362451559942453658464959, −6.70829628609736489469726496074, −5.85298835130201725932666686958, −5.50793755064970537782897699538, −5.10345850241449878208474053042, −4.90311304787222042896478234940, −3.80548835742286292236677637623, −3.74097983086957311898641067849, −2.83572130224359276673610014690, −2.71596232140089496944594888065, −1.51428737876706291554884790017, −1.44898043231751531467226703938, 0, 0,
1.44898043231751531467226703938, 1.51428737876706291554884790017, 2.71596232140089496944594888065, 2.83572130224359276673610014690, 3.74097983086957311898641067849, 3.80548835742286292236677637623, 4.90311304787222042896478234940, 5.10345850241449878208474053042, 5.50793755064970537782897699538, 5.85298835130201725932666686958, 6.70829628609736489469726496074, 6.92346362451559942453658464959, 7.58623751188550053148364349374, 7.62539202873709958105730203929, 8.328959020043083457743168771166, 8.556180195376830616700194636366, 9.270608277120423900144699495153, 9.458873460454900163766899739430
