| L(s) = 1 | + 8·11-s − 36·17-s + 24·19-s + 46·25-s − 28·41-s + 56·43-s + 34·49-s + 104·59-s − 8·67-s − 132·73-s − 280·83-s − 60·89-s − 28·97-s − 312·107-s − 196·113-s − 194·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 142·169-s + 173-s + ⋯ |
| L(s) = 1 | + 8/11·11-s − 2.11·17-s + 1.26·19-s + 1.83·25-s − 0.682·41-s + 1.30·43-s + 0.693·49-s + 1.76·59-s − 0.119·67-s − 1.80·73-s − 3.37·83-s − 0.674·89-s − 0.288·97-s − 2.91·107-s − 1.73·113-s − 1.60·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.840·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.896187529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896187529\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 142 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 542 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1486 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 898 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1838 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4162 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1262 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 718 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6946 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12226 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 140 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
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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
−9.060714480967077384986164550781, −8.694626857682880956613881407611, −8.464354742660311588112614134130, −7.74418809318718047619012766589, −7.52813796594117754116480388757, −6.84627407533303238271333037674, −6.70479724491853084954998419344, −6.64417677792203423865510263904, −5.78782984936074719195243181819, −5.43801382566198138191347627896, −5.17486621881442857079067037867, −4.41014792053344680013675243790, −4.28028029658507749168266796459, −3.87450988093561287763792746996, −3.16624179055811715560990854326, −2.56994262751540823952920347494, −2.53481561775230157115243610200, −1.40685001618498820329014534203, −1.27980813534481900663269430524, −0.34989281455098309243453279081,
0.34989281455098309243453279081, 1.27980813534481900663269430524, 1.40685001618498820329014534203, 2.53481561775230157115243610200, 2.56994262751540823952920347494, 3.16624179055811715560990854326, 3.87450988093561287763792746996, 4.28028029658507749168266796459, 4.41014792053344680013675243790, 5.17486621881442857079067037867, 5.43801382566198138191347627896, 5.78782984936074719195243181819, 6.64417677792203423865510263904, 6.70479724491853084954998419344, 6.84627407533303238271333037674, 7.52813796594117754116480388757, 7.74418809318718047619012766589, 8.464354742660311588112614134130, 8.694626857682880956613881407611, 9.060714480967077384986164550781
