| L(s) = 1 | + 2·4-s + 9-s + 16-s + 4·19-s − 4·31-s + 2·36-s − 49-s − 4·61-s − 2·64-s + 8·76-s − 2·79-s − 2·109-s + 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 4·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s + 9-s + 16-s + 4·19-s − 4·31-s + 2·36-s − 49-s − 4·61-s − 2·64-s + 8·76-s − 2·79-s − 2·109-s + 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 4·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.815299436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.815299436\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.48798319002898440444146479794, −6.16760697365510773660104628599, −6.03535144523682538884445128663, −5.74020949622263983658758002605, −5.73798197336295451693772336730, −5.63707782312064713724533046933, −5.08224361122067076829324263509, −5.07261599624894089950998795339, −4.99893724517073529666204536598, −4.50758924548544409562797433188, −4.47510087751680836832660732501, −4.04966433195771582835647299868, −3.92876626880542804710367814715, −3.46283218285290825074213414428, −3.42844506330129041196371243280, −3.17714460936252941051868439030, −3.00507688105172873945131661509, −2.83792109104216925563622807242, −2.46580081951396923268192326839, −2.24627510075681699788047842975, −1.68563572102360427890900640987, −1.58968975608278260815546717874, −1.55249092406693099271471259034, −1.39691385612135329514453256174, −0.61142252351547762562900073435,
0.61142252351547762562900073435, 1.39691385612135329514453256174, 1.55249092406693099271471259034, 1.58968975608278260815546717874, 1.68563572102360427890900640987, 2.24627510075681699788047842975, 2.46580081951396923268192326839, 2.83792109104216925563622807242, 3.00507688105172873945131661509, 3.17714460936252941051868439030, 3.42844506330129041196371243280, 3.46283218285290825074213414428, 3.92876626880542804710367814715, 4.04966433195771582835647299868, 4.47510087751680836832660732501, 4.50758924548544409562797433188, 4.99893724517073529666204536598, 5.07261599624894089950998795339, 5.08224361122067076829324263509, 5.63707782312064713724533046933, 5.73798197336295451693772336730, 5.74020949622263983658758002605, 6.03535144523682538884445128663, 6.16760697365510773660104628599, 6.48798319002898440444146479794
