| L(s) = 1 | + 3-s + 4-s + 4·5-s − 8·7-s + 3·9-s + 12-s + 4·15-s + 16-s + 6·17-s + 4·20-s − 8·21-s + 10·25-s + 8·27-s − 8·28-s − 32·35-s + 3·36-s − 4·37-s − 24·41-s − 4·43-s + 12·45-s − 18·47-s + 48-s + 34·49-s + 6·51-s − 12·59-s + 4·60-s − 24·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.78·5-s − 3.02·7-s + 9-s + 0.288·12-s + 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.894·20-s − 1.74·21-s + 2·25-s + 1.53·27-s − 1.51·28-s − 5.40·35-s + 1/2·36-s − 0.657·37-s − 3.74·41-s − 0.609·43-s + 1.78·45-s − 2.62·47-s + 0.144·48-s + 34/7·49-s + 0.840·51-s − 1.56·59-s + 0.516·60-s − 3.02·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422018991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422018991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - T^{2} + 264 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 3960 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4134 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 150 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 265 T^{2} + 32736 T^{4} - 265 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07184794099349296693698259091, −9.970339824414030403122522356040, −9.592083117807769712215864373864, −9.574581561373486710942169590745, −9.274234105409606876667414574047, −8.675881314756184674941142998726, −8.635526363109768257033067448781, −8.354251635499284084465239115798, −7.65702002748354059575768075344, −7.53853794170120613678327812209, −6.99914200406531654672298149068, −6.66631235715714539946724161788, −6.61999853750819441477048005630, −6.43395514024750228163581171401, −6.01296208264307035323013516221, −5.72745510780103229530670577421, −5.09420919108976956957106794419, −5.08898604836189796432940207717, −4.47114510218065475631758746190, −3.53528712972967195973977795790, −3.46983025312184155536101997926, −3.07061117435097738365517582810, −2.91819286725912666469207734611, −1.97639919723049730284999905793, −1.55410476365347686804164752499,
1.55410476365347686804164752499, 1.97639919723049730284999905793, 2.91819286725912666469207734611, 3.07061117435097738365517582810, 3.46983025312184155536101997926, 3.53528712972967195973977795790, 4.47114510218065475631758746190, 5.08898604836189796432940207717, 5.09420919108976956957106794419, 5.72745510780103229530670577421, 6.01296208264307035323013516221, 6.43395514024750228163581171401, 6.61999853750819441477048005630, 6.66631235715714539946724161788, 6.99914200406531654672298149068, 7.53853794170120613678327812209, 7.65702002748354059575768075344, 8.354251635499284084465239115798, 8.635526363109768257033067448781, 8.675881314756184674941142998726, 9.274234105409606876667414574047, 9.574581561373486710942169590745, 9.592083117807769712215864373864, 9.970339824414030403122522356040, 10.07184794099349296693698259091
