L(s) = 1 | + 2·4-s + 8·13-s + 2·19-s − 4·25-s + 14·31-s + 16·37-s − 2·43-s + 16·52-s − 10·61-s − 8·64-s + 4·67-s + 2·73-s + 4·76-s − 8·79-s + 2·97-s − 8·100-s − 4·103-s − 2·109-s + 2·121-s + 28·124-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 4-s + 2.21·13-s + 0.458·19-s − 4/5·25-s + 2.51·31-s + 2.63·37-s − 0.304·43-s + 2.21·52-s − 1.28·61-s − 64-s + 0.488·67-s + 0.234·73-s + 0.458·76-s − 0.900·79-s + 0.203·97-s − 4/5·100-s − 0.394·103-s − 0.191·109-s + 2/11·121-s + 2.51·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.684094338\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.684094338\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 172 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−9.704576184006212401406534391892, −9.659799394167776585919234634684, −8.858517335179421661812818226589, −8.658548834767622359693356785920, −8.171208078533122140812875556216, −7.75619633321026076769561160862, −7.54248163045900977711521273669, −6.84764811184588095689576540153, −6.35256073278370905685106600630, −6.22286502589896433105920577290, −5.96525296975031089119311462887, −5.31807735862081517667126900023, −4.65185158550124614932619315480, −4.20073293352396634439772356827, −3.82589597855794928672562641004, −3.01162930119869255069615843900, −2.85228745878207005705032591050, −2.10649687865595426246430127606, −1.38682798543293263600778396077, −0.876824984314849652791927170507,
0.876824984314849652791927170507, 1.38682798543293263600778396077, 2.10649687865595426246430127606, 2.85228745878207005705032591050, 3.01162930119869255069615843900, 3.82589597855794928672562641004, 4.20073293352396634439772356827, 4.65185158550124614932619315480, 5.31807735862081517667126900023, 5.96525296975031089119311462887, 6.22286502589896433105920577290, 6.35256073278370905685106600630, 6.84764811184588095689576540153, 7.54248163045900977711521273669, 7.75619633321026076769561160862, 8.171208078533122140812875556216, 8.658548834767622359693356785920, 8.858517335179421661812818226589, 9.659799394167776585919234634684, 9.704576184006212401406534391892