| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 10-s − 4·13-s + 15-s − 16-s − 6·17-s + 8·19-s − 24-s − 4·26-s − 27-s + 12·29-s + 30-s − 4·31-s − 6·34-s + 10·37-s + 8·38-s − 4·39-s − 40-s + 12·41-s − 8·43-s − 48-s − 6·51-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.316·10-s − 1.10·13-s + 0.258·15-s − 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 2.22·29-s + 0.182·30-s − 0.718·31-s − 1.02·34-s + 1.64·37-s + 1.29·38-s − 0.640·39-s − 0.158·40-s + 1.87·41-s − 1.21·43-s − 0.144·48-s − 0.840·51-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.649711969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.649711969\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.728497547498058533735334735836, −9.338080994032944608485989654332, −8.894303070404387397383973723733, −8.650018660748479609457998307355, −8.038269448391604779238755136992, −7.68254853713483531215514148454, −7.29740415629466908276274095435, −6.85182584048955383454710540330, −6.42931842579313352490323337702, −5.88876688056146524080617081376, −5.64881373260640418407978758687, −4.84708268021140575327901544115, −4.79675840640872542536590620437, −4.30649302295543704883241294508, −3.74678196710062164601131419384, −2.97515717724974212137352797841, −2.80844703277181146001516374526, −2.34334065828303893841005143162, −1.56102080713007626414566578129, −0.65780761910369101776184919206,
0.65780761910369101776184919206, 1.56102080713007626414566578129, 2.34334065828303893841005143162, 2.80844703277181146001516374526, 2.97515717724974212137352797841, 3.74678196710062164601131419384, 4.30649302295543704883241294508, 4.79675840640872542536590620437, 4.84708268021140575327901544115, 5.64881373260640418407978758687, 5.88876688056146524080617081376, 6.42931842579313352490323337702, 6.85182584048955383454710540330, 7.29740415629466908276274095435, 7.68254853713483531215514148454, 8.038269448391604779238755136992, 8.650018660748479609457998307355, 8.894303070404387397383973723733, 9.338080994032944608485989654332, 9.728497547498058533735334735836
