| L(s) = 1 | + 4·2-s − 2·3-s + 8·4-s + 4·5-s − 8·6-s + 8·8-s + 3·9-s + 16·10-s − 16·12-s − 8·15-s − 4·16-s + 12·17-s + 12·18-s + 32·20-s − 16·24-s + 11·25-s − 4·27-s − 4·29-s − 32·30-s − 32·32-s + 48·34-s + 24·36-s + 2·37-s + 32·40-s + 2·43-s + 12·45-s − 16·47-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 4·4-s + 1.78·5-s − 3.26·6-s + 2.82·8-s + 9-s + 5.05·10-s − 4.61·12-s − 2.06·15-s − 16-s + 2.91·17-s + 2.82·18-s + 7.15·20-s − 3.26·24-s + 11/5·25-s − 0.769·27-s − 0.742·29-s − 5.84·30-s − 5.65·32-s + 8.23·34-s + 4·36-s + 0.328·37-s + 5.05·40-s + 0.304·43-s + 1.78·45-s − 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.925930624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.925930624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
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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
−11.53404899071342143673972193440, −11.31964077809128755766758793830, −10.66518176202523632624171352051, −10.07614116417574186181496133968, −9.777626087947438429350619065172, −9.476660412762067745707152239415, −8.696753834975057074824724346127, −7.969692709288310492536008623657, −7.00172992126185058755452452244, −6.95425659078922914311067993793, −6.10830894406878056202091453064, −5.87383952946041732115373603645, −5.54892767092728099639388658978, −5.34026607346045504613598208763, −4.74896060628684584323173366671, −4.27497789850775790366433240614, −3.33671717360574232261915277963, −3.18473962013872326817762539604, −2.22851511724058885799394735378, −1.36945748312626220732479902083,
1.36945748312626220732479902083, 2.22851511724058885799394735378, 3.18473962013872326817762539604, 3.33671717360574232261915277963, 4.27497789850775790366433240614, 4.74896060628684584323173366671, 5.34026607346045504613598208763, 5.54892767092728099639388658978, 5.87383952946041732115373603645, 6.10830894406878056202091453064, 6.95425659078922914311067993793, 7.00172992126185058755452452244, 7.969692709288310492536008623657, 8.696753834975057074824724346127, 9.476660412762067745707152239415, 9.777626087947438429350619065172, 10.07614116417574186181496133968, 10.66518176202523632624171352051, 11.31964077809128755766758793830, 11.53404899071342143673972193440
