| L(s) = 1 | − 9·9-s − 72·11-s − 200·19-s + 468·29-s + 32·31-s + 180·41-s + 622·49-s − 1.36e3·59-s + 844·61-s + 720·71-s + 1.02e3·79-s + 81·81-s + 1.26e3·89-s + 648·99-s + 1.11e3·101-s − 3.24e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.29e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.97·11-s − 2.41·19-s + 2.99·29-s + 0.185·31-s + 0.685·41-s + 1.81·49-s − 3.01·59-s + 1.77·61-s + 1.20·71-s + 1.45·79-s + 1/9·81-s + 1.50·89-s + 0.657·99-s + 1.09·101-s − 2.85·109-s + 0.921·121-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 + 1.95·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.011802968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011802968\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9502 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 21022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 684 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 777358 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 714430 T^{2} + p^{6} T^{4} \) |
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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
−9.381881892869933960182162013941, −9.315585791559626645718932387939, −8.475333081674972906126037336195, −8.433470868779123943136003481536, −7.945042911459347257143714997860, −7.81782100322514914359738800832, −6.89505289257423392843508682470, −6.79196205248417130059729620231, −6.17198739622586690176000977699, −5.90177555109662382626326301929, −5.25821360304560886486595253284, −4.91573899118381531620797614928, −4.34449442111824018043152217349, −4.17885844787549614717750037822, −3.14021343648466992500253467655, −2.90007852801176338547779338095, −2.24664503001311452911475015496, −2.03677823919200040711537609870, −0.801574878833457261017641122618, −0.44215439147944638013897891276,
0.44215439147944638013897891276, 0.801574878833457261017641122618, 2.03677823919200040711537609870, 2.24664503001311452911475015496, 2.90007852801176338547779338095, 3.14021343648466992500253467655, 4.17885844787549614717750037822, 4.34449442111824018043152217349, 4.91573899118381531620797614928, 5.25821360304560886486595253284, 5.90177555109662382626326301929, 6.17198739622586690176000977699, 6.79196205248417130059729620231, 6.89505289257423392843508682470, 7.81782100322514914359738800832, 7.945042911459347257143714997860, 8.433470868779123943136003481536, 8.475333081674972906126037336195, 9.315585791559626645718932387939, 9.381881892869933960182162013941
