| L(s) = 1 | + 60·11-s + 56·19-s − 420·29-s − 8·31-s + 480·41-s + 682·49-s + 900·59-s − 332·61-s − 2.04e3·71-s + 1.83e3·79-s + 840·89-s + 900·101-s + 3.40e3·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.37e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.64·11-s + 0.676·19-s − 2.68·29-s − 0.0463·31-s + 1.82·41-s + 1.98·49-s + 1.98·59-s − 0.696·61-s − 3.40·71-s + 2.60·79-s + 1.00·89-s + 0.886·101-s + 2.99·109-s + 0.0285·121-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 + 1.99·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.520142475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520142475\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4378 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1726 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 61306 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 140518 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 193246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 450 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 222938 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1020 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 715534 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 916 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 156026 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 540098 T^{2} + p^{6} T^{4} \) |
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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
−9.862048838099089684281713303064, −9.327477117999820591859985928513, −9.077458509640400681824568308323, −8.984757123786402427642232785285, −8.303989356516527336828144194491, −7.64521413829382144285755038282, −7.29274695838758324888561995994, −7.21412527715072580097826691702, −6.39292657444229191763917115297, −6.05734782291819883945001197779, −5.63442383737230546666706575152, −5.22743037597371884508436182095, −4.40374737342277227871344860888, −4.11666844502765768121624199982, −3.61601024255644277325011327017, −3.18367518960115744679590450596, −2.28149160136605896831341064411, −1.85831384695908548445332944375, −1.08753953328986409167968713206, −0.55303527308519681293374018256,
0.55303527308519681293374018256, 1.08753953328986409167968713206, 1.85831384695908548445332944375, 2.28149160136605896831341064411, 3.18367518960115744679590450596, 3.61601024255644277325011327017, 4.11666844502765768121624199982, 4.40374737342277227871344860888, 5.22743037597371884508436182095, 5.63442383737230546666706575152, 6.05734782291819883945001197779, 6.39292657444229191763917115297, 7.21412527715072580097826691702, 7.29274695838758324888561995994, 7.64521413829382144285755038282, 8.303989356516527336828144194491, 8.984757123786402427642232785285, 9.077458509640400681824568308323, 9.327477117999820591859985928513, 9.862048838099089684281713303064
