| L(s) = 1 | − 5·5-s + 35·9-s − 22·11-s + 136·19-s − 100·25-s − 520·29-s + 350·31-s − 760·41-s − 175·45-s + 610·49-s + 110·55-s + 286·59-s + 1.35e3·61-s + 2.07e3·71-s − 436·79-s + 496·81-s − 2.55e3·89-s − 680·95-s − 770·99-s + 1.27e3·101-s + 284·109-s + 363·121-s + 1.12e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.29·9-s − 0.603·11-s + 1.64·19-s − 4/5·25-s − 3.32·29-s + 2.02·31-s − 2.89·41-s − 0.579·45-s + 1.77·49-s + 0.269·55-s + 0.631·59-s + 2.83·61-s + 3.46·71-s − 0.620·79-s + 0.680·81-s − 3.04·89-s − 0.734·95-s − 0.781·99-s + 1.25·101-s + 0.249·109-s + 3/11·121-s + 0.804·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.035427755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.035427755\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 35 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9142 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10483 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 260 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 175 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 72407 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 380 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 114546 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 92250 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 143 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 676 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 323347 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1035 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 668290 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 218 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 568330 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1279 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1230095 T^{2} + p^{6} T^{4} \) |
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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
−11.95074739280655888122812547081, −11.58639949252771081875024192292, −11.30284408933867571540769753388, −10.56376511680353018762772406797, −9.897713089129453427350787820338, −9.859277426808712231543834332675, −9.347860888262493976750365016387, −8.376785139584930230905656127960, −8.184744302757415598630857206314, −7.43236208204780404458438339723, −7.12897220904777658051349220430, −6.71616314121717887024652794718, −5.58230306706859417250584546049, −5.42010855150368838168576024934, −4.66051454224359381589732379373, −3.71069324037631876847393722610, −3.66923080155452296311962486922, −2.43516102729658480829965200191, −1.63036085856316864720049048689, −0.61625561810630477085879647291,
0.61625561810630477085879647291, 1.63036085856316864720049048689, 2.43516102729658480829965200191, 3.66923080155452296311962486922, 3.71069324037631876847393722610, 4.66051454224359381589732379373, 5.42010855150368838168576024934, 5.58230306706859417250584546049, 6.71616314121717887024652794718, 7.12897220904777658051349220430, 7.43236208204780404458438339723, 8.184744302757415598630857206314, 8.376785139584930230905656127960, 9.347860888262493976750365016387, 9.859277426808712231543834332675, 9.897713089129453427350787820338, 10.56376511680353018762772406797, 11.30284408933867571540769753388, 11.58639949252771081875024192292, 11.95074739280655888122812547081
