| L(s) = 1 | + 6·9-s − 50·25-s − 144·29-s + 72·41-s + 20·49-s + 296·61-s + 27·81-s + 72·89-s − 144·101-s − 104·109-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2·25-s − 4.96·29-s + 1.75·41-s + 0.408·49-s + 4.85·61-s + 1/3·81-s + 0.808·89-s − 1.42·101-s − 0.954·109-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.165·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7781233972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7781233972\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 178 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 4942 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5990 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 7250 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 718 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9362 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 5666 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 13634 T^{2} + p^{4} T^{4} )^{2} \) |
| 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
−7.02296948193864081245242545617, −6.79254761920747454071435528452, −6.63036196107310203467228732836, −6.30055371753355314058630773945, −5.88177630943575908524729801212, −5.72957608371297792952821657852, −5.60467604366984907486545822971, −5.59014390618975757554875717084, −5.23686464897032305053045865667, −4.96666397511866811299819884950, −4.51550869231076721634850722952, −4.33392738028215068882207461963, −3.95112752236971144011578563588, −3.88193388633757651489564683164, −3.82634061070980053772342529961, −3.39854392845080213094766336230, −3.22160525526512845722656753553, −2.65792896148239648001035362452, −2.33010697967007716358992504207, −2.04674659355823915829253314929, −1.84753715509971178583210598063, −1.76423393508129699424519992716, −0.952697880073195944485700767324, −0.78196805635026846277212599965, −0.14230542934188463885537818562,
0.14230542934188463885537818562, 0.78196805635026846277212599965, 0.952697880073195944485700767324, 1.76423393508129699424519992716, 1.84753715509971178583210598063, 2.04674659355823915829253314929, 2.33010697967007716358992504207, 2.65792896148239648001035362452, 3.22160525526512845722656753553, 3.39854392845080213094766336230, 3.82634061070980053772342529961, 3.88193388633757651489564683164, 3.95112752236971144011578563588, 4.33392738028215068882207461963, 4.51550869231076721634850722952, 4.96666397511866811299819884950, 5.23686464897032305053045865667, 5.59014390618975757554875717084, 5.60467604366984907486545822971, 5.72957608371297792952821657852, 5.88177630943575908524729801212, 6.30055371753355314058630773945, 6.63036196107310203467228732836, 6.79254761920747454071435528452, 7.02296948193864081245242545617
