| L(s) = 1 | + 2·5-s − 4·7-s − 2·11-s + 4·13-s − 4·19-s + 3·25-s + 8·31-s − 8·35-s + 4·37-s − 4·43-s − 2·49-s + 12·53-s − 4·55-s + 4·61-s + 8·65-s − 16·67-s + 4·73-s + 8·77-s + 20·79-s + 24·83-s + 12·89-s − 16·91-s − 8·95-s − 20·97-s − 16·103-s + 24·107-s − 20·109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.603·11-s + 1.10·13-s − 0.917·19-s + 3/5·25-s + 1.43·31-s − 1.35·35-s + 0.657·37-s − 0.609·43-s − 2/7·49-s + 1.64·53-s − 0.539·55-s + 0.512·61-s + 0.992·65-s − 1.95·67-s + 0.468·73-s + 0.911·77-s + 2.25·79-s + 2.63·83-s + 1.27·89-s − 1.67·91-s − 0.820·95-s − 2.03·97-s − 1.57·103-s + 2.32·107-s − 1.91·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.232634821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.232634821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.026407708054951337820521172458, −7.78594280718458666254744357949, −7.16498047217035039555164161798, −6.85420855449600597353886994838, −6.47747047049014084290698821884, −6.38229139599154323866638001394, −5.96454399954361730152367859326, −5.80699869250567149778161706744, −5.18140861575531995253965759736, −5.01986966619532391648883280063, −4.33772352813683367756699412549, −4.21769149274301011170668161916, −3.52112900791057735897113384396, −3.34956595105040784799329489917, −2.83079560782821657842389682170, −2.60302676456179642555635389026, −1.92464104324579127867055050797, −1.73367994208127801224848440072, −0.73255494986985676588407775757, −0.59428919456649012438210385244,
0.59428919456649012438210385244, 0.73255494986985676588407775757, 1.73367994208127801224848440072, 1.92464104324579127867055050797, 2.60302676456179642555635389026, 2.83079560782821657842389682170, 3.34956595105040784799329489917, 3.52112900791057735897113384396, 4.21769149274301011170668161916, 4.33772352813683367756699412549, 5.01986966619532391648883280063, 5.18140861575531995253965759736, 5.80699869250567149778161706744, 5.96454399954361730152367859326, 6.38229139599154323866638001394, 6.47747047049014084290698821884, 6.85420855449600597353886994838, 7.16498047217035039555164161798, 7.78594280718458666254744357949, 8.026407708054951337820521172458
