| L(s) = 1 | + 2·2-s − 4-s + 4·5-s − 6·7-s − 8·8-s + 8·10-s + 6·11-s − 12·14-s − 7·16-s + 2·17-s − 4·20-s + 12·22-s − 2·23-s + 11·25-s + 6·28-s + 6·29-s − 2·31-s + 14·32-s + 4·34-s − 24·35-s + 6·37-s − 32·40-s + 6·41-s − 24·43-s − 6·44-s − 4·46-s + 18·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 1.78·5-s − 2.26·7-s − 2.82·8-s + 2.52·10-s + 1.80·11-s − 3.20·14-s − 7/4·16-s + 0.485·17-s − 0.894·20-s + 2.55·22-s − 0.417·23-s + 11/5·25-s + 1.13·28-s + 1.11·29-s − 0.359·31-s + 2.47·32-s + 0.685·34-s − 4.05·35-s + 0.986·37-s − 5.05·40-s + 0.937·41-s − 3.65·43-s − 0.904·44-s − 0.589·46-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.754486606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754486606\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 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
−10.01421227292562997425368914585, −10.01417999387952348108139911508, −9.764696328332092371332326155532, −9.405015645985347767999405015077, −8.968064603115670831350823445251, −8.722706098322371824885722428154, −8.195352989633790082052462822372, −7.11398182546144966364178912960, −6.55588023340281853038562674456, −6.41614962037159127408575775289, −6.11153285809745545616333007614, −5.72387926305962108519581371330, −5.21416372263119210620104416251, −4.64164419894372938613623123926, −4.15317359968799817114118396827, −3.55923127939053241761960310268, −3.12636817055897769903891892752, −2.87962475821772893579073204267, −1.77709805435321349525526814304, −0.69290661148862701190294375233,
0.69290661148862701190294375233, 1.77709805435321349525526814304, 2.87962475821772893579073204267, 3.12636817055897769903891892752, 3.55923127939053241761960310268, 4.15317359968799817114118396827, 4.64164419894372938613623123926, 5.21416372263119210620104416251, 5.72387926305962108519581371330, 6.11153285809745545616333007614, 6.41614962037159127408575775289, 6.55588023340281853038562674456, 7.11398182546144966364178912960, 8.195352989633790082052462822372, 8.722706098322371824885722428154, 8.968064603115670831350823445251, 9.405015645985347767999405015077, 9.764696328332092371332326155532, 10.01417999387952348108139911508, 10.01421227292562997425368914585
