| L(s) = 1 | + 2·2-s + 2·4-s − 6·7-s − 24·11-s − 24·13-s − 12·14-s − 4·16-s − 24·17-s − 48·22-s − 6·23-s − 48·26-s − 12·28-s − 16·31-s − 8·32-s − 48·34-s − 96·37-s + 96·41-s − 54·43-s − 48·44-s − 12·46-s − 54·47-s + 18·49-s − 48·52-s + 24·53-s + 64·61-s − 32·62-s − 8·64-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 6/7·7-s − 2.18·11-s − 1.84·13-s − 6/7·14-s − 1/4·16-s − 1.41·17-s − 2.18·22-s − 0.260·23-s − 1.84·26-s − 3/7·28-s − 0.516·31-s − 1/4·32-s − 1.41·34-s − 2.59·37-s + 2.34·41-s − 1.25·43-s − 1.09·44-s − 0.260·46-s − 1.14·47-s + 0.367·49-s − 0.923·52-s + 0.452·53-s + 1.04·61-s − 0.516·62-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2543519927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2543519927\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 186 T + 17298 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.45978368462385252032605115576, −10.44294953053361651290031001357, −10.44045677273259063978480810578, −9.760960734492452382314169658146, −9.564319432652994993661748681734, −8.662657052204385408560293276179, −8.531020441407849239558909366870, −7.63620326212253654900255948705, −7.39623409825557126083905220068, −6.88165710361960955613939425367, −6.46147206179342856635772207486, −5.61385385054849307645092436142, −5.43532237680396697126432624571, −4.72762904327044007051422065485, −4.59503965952966961387836556509, −3.59399985154746245982779657036, −3.12150638235640145834482265512, −2.27355510390479788293446068657, −2.26501707871681033292559392503, −0.16075624130832180634033572950,
0.16075624130832180634033572950, 2.26501707871681033292559392503, 2.27355510390479788293446068657, 3.12150638235640145834482265512, 3.59399985154746245982779657036, 4.59503965952966961387836556509, 4.72762904327044007051422065485, 5.43532237680396697126432624571, 5.61385385054849307645092436142, 6.46147206179342856635772207486, 6.88165710361960955613939425367, 7.39623409825557126083905220068, 7.63620326212253654900255948705, 8.531020441407849239558909366870, 8.662657052204385408560293276179, 9.564319432652994993661748681734, 9.760960734492452382314169658146, 10.44045677273259063978480810578, 10.44294953053361651290031001357, 11.45978368462385252032605115576
