| L(s) = 1 | + 10·5-s − 9·9-s + 56·11-s + 64·19-s − 25·25-s + 476·29-s + 360·31-s + 844·41-s − 90·45-s + 670·49-s + 560·55-s − 1.60e3·59-s − 716·61-s + 128·71-s − 1.86e3·79-s + 81·81-s + 2.29e3·89-s + 640·95-s − 504·99-s − 2.58e3·101-s − 1.93e3·109-s − 310·121-s − 1.50e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s + 1.53·11-s + 0.772·19-s − 1/5·25-s + 3.04·29-s + 2.08·31-s + 3.21·41-s − 0.298·45-s + 1.95·49-s + 1.37·55-s − 3.54·59-s − 1.50·61-s + 0.213·71-s − 2.65·79-s + 1/9·81-s + 2.72·89-s + 0.691·95-s − 0.511·99-s − 2.54·101-s − 1.69·109-s − 0.232·121-s − 1.07·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.812093381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.812093381\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4138 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1838 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 23550 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 99706 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82838 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 249354 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 804 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 179930 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 754930 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 932 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 525690 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1146 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1146370 T^{2} + p^{6} T^{4} \) |
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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
−11.79375861184102756700748526799, −11.76327068350993535229702077340, −10.79591764249374451171075920018, −10.52825044424822434810864330674, −9.958149482380564699630390111721, −9.477212736153836855739827678743, −9.005167162089953398063169195783, −8.781598239961430720847544262961, −7.80560201457035987215768291110, −7.66685553852563841546974788215, −6.58661833916476965887301379413, −6.35512758739217660217845337291, −6.00837591776011731007855101178, −5.24330595345181253775667435599, −4.36599882740992326790353609372, −4.21969795498280073977727278028, −2.80950935318641307171717475215, −2.77595394874203796839200201053, −1.38914783662101250070884485951, −0.916639085499223016696018984617,
0.916639085499223016696018984617, 1.38914783662101250070884485951, 2.77595394874203796839200201053, 2.80950935318641307171717475215, 4.21969795498280073977727278028, 4.36599882740992326790353609372, 5.24330595345181253775667435599, 6.00837591776011731007855101178, 6.35512758739217660217845337291, 6.58661833916476965887301379413, 7.66685553852563841546974788215, 7.80560201457035987215768291110, 8.781598239961430720847544262961, 9.005167162089953398063169195783, 9.477212736153836855739827678743, 9.958149482380564699630390111721, 10.52825044424822434810864330674, 10.79591764249374451171075920018, 11.76327068350993535229702077340, 11.79375861184102756700748526799
