| L(s) = 1 | − 12·7-s + 50·9-s + 52·17-s + 156·23-s − 25·25-s − 216·31-s − 44·41-s − 1.02e3·47-s − 578·49-s − 600·63-s − 824·71-s + 1.75e3·73-s + 1.20e3·79-s + 1.77e3·81-s + 300·89-s + 772·97-s + 1.19e3·103-s + 3.12e3·113-s − 624·119-s + 1.63e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.60e3·153-s + ⋯ |
| L(s) = 1 | − 0.647·7-s + 1.85·9-s + 0.741·17-s + 1.41·23-s − 1/5·25-s − 1.25·31-s − 0.167·41-s − 3.19·47-s − 1.68·49-s − 1.19·63-s − 1.37·71-s + 2.81·73-s + 1.70·79-s + 2.42·81-s + 0.357·89-s + 0.808·97-s + 1.14·103-s + 2.60·113-s − 0.480·119-s + 1.23·121-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 + 1.37·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.313081129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.313081129\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1638 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46278 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30550 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 514 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 297750 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 160758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 185638 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 585650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 878 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 600 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1064050 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−9.502840790716678306870017170525, −9.379882434636669955351346149267, −8.648867879379876927101683888800, −8.324399933382693724596262214049, −7.61063701121957513711166948380, −7.60701935564063159120556809079, −6.95888381870986730612657390245, −6.70782261965001786869428689032, −6.31868363154200435459794213377, −5.82575371851561993031743552373, −5.03053531508750380963378540030, −4.91936563217609242711024672185, −4.48623018539530115482068681255, −3.70643185142289997713092646597, −3.30123780713131655900810561388, −3.19947598367517014474693879066, −1.94573033343062993260221654129, −1.84648132656004475961690105100, −1.04467332713503474572271910760, −0.47795396462831431387301314890,
0.47795396462831431387301314890, 1.04467332713503474572271910760, 1.84648132656004475961690105100, 1.94573033343062993260221654129, 3.19947598367517014474693879066, 3.30123780713131655900810561388, 3.70643185142289997713092646597, 4.48623018539530115482068681255, 4.91936563217609242711024672185, 5.03053531508750380963378540030, 5.82575371851561993031743552373, 6.31868363154200435459794213377, 6.70782261965001786869428689032, 6.95888381870986730612657390245, 7.60701935564063159120556809079, 7.61063701121957513711166948380, 8.324399933382693724596262214049, 8.648867879379876927101683888800, 9.379882434636669955351346149267, 9.502840790716678306870017170525
