| L(s) = 1 | − 2·2-s + 2·4-s − 4·7-s + 16·11-s − 6·13-s + 8·14-s − 4·16-s + 14·17-s − 32·22-s − 4·23-s + 12·26-s − 8·28-s + 104·31-s + 8·32-s − 28·34-s + 6·37-s + 16·41-s + 84·43-s + 32·44-s + 8·46-s − 36·47-s + 8·49-s − 12·52-s + 106·53-s − 96·61-s − 208·62-s − 8·64-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 4/7·7-s + 1.45·11-s − 0.461·13-s + 4/7·14-s − 1/4·16-s + 0.823·17-s − 1.45·22-s − 0.173·23-s + 6/13·26-s − 2/7·28-s + 3.35·31-s + 1/4·32-s − 0.823·34-s + 6/37·37-s + 0.390·41-s + 1.95·43-s + 8/11·44-s + 4/23·46-s − 0.765·47-s + 8/49·49-s − 0.230·52-s + 2·53-s − 1.57·61-s − 3.35·62-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.498458307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498458307\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 124 T + 7688 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 94 T + 4418 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.89973485394423228991887064032, −10.57844523986801661046652745130, −9.969914673082683395379026664713, −9.819717063780263940831006374941, −9.272144050406899132801653460822, −9.039023448371954811076884927066, −8.341407801644792967206259845260, −8.087741701166242533511368086737, −7.44660913166022219059097195682, −7.06823533805341154226414318499, −6.44043667335696521449679087246, −6.16169616131113059014241265875, −5.62365474716488421471665282659, −4.64117741429800749792893083448, −4.36991418706128983134629750208, −3.60765048500981809028245086443, −2.91634252430250243227674852982, −2.29002443351873871171991710162, −1.21034372367772908446075604506, −0.71516924696457514430741542876,
0.71516924696457514430741542876, 1.21034372367772908446075604506, 2.29002443351873871171991710162, 2.91634252430250243227674852982, 3.60765048500981809028245086443, 4.36991418706128983134629750208, 4.64117741429800749792893083448, 5.62365474716488421471665282659, 6.16169616131113059014241265875, 6.44043667335696521449679087246, 7.06823533805341154226414318499, 7.44660913166022219059097195682, 8.087741701166242533511368086737, 8.341407801644792967206259845260, 9.039023448371954811076884927066, 9.272144050406899132801653460822, 9.819717063780263940831006374941, 9.969914673082683395379026664713, 10.57844523986801661046652745130, 10.89973485394423228991887064032
