| L(s) = 1 | + 2-s + 4-s + 1.24·5-s + 8-s + 9-s + 1.24·10-s − 1.80·13-s + 16-s − 0.445·17-s + 18-s + 1.24·20-s + 0.554·25-s − 1.80·26-s + 32-s − 0.445·34-s + 36-s − 1.80·37-s + 1.24·40-s − 1.80·41-s + 1.24·45-s + 49-s + 0.554·50-s − 1.80·52-s − 0.445·53-s − 0.445·61-s + 64-s − 2.24·65-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.24·5-s + 8-s + 9-s + 1.24·10-s − 1.80·13-s + 16-s − 0.445·17-s + 18-s + 1.24·20-s + 0.554·25-s − 1.80·26-s + 32-s − 0.445·34-s + 36-s − 1.80·37-s + 1.24·40-s − 1.80·41-s + 1.24·45-s + 49-s + 0.554·50-s − 1.80·52-s − 0.445·53-s − 0.445·61-s + 64-s − 2.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.004681667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.004681667\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.911241264865647419466454365586, −7.74965213848326785022799319923, −6.99549741217795448239305223620, −6.61642633441396966534873934476, −5.57059668384447078051761608503, −5.03023761593222638131806825818, −4.35923000731639677080293535208, −3.24311040591428083147352743781, −2.22939093840673703378916003036, −1.69125022609278132803550252304,
1.69125022609278132803550252304, 2.22939093840673703378916003036, 3.24311040591428083147352743781, 4.35923000731639677080293535208, 5.03023761593222638131806825818, 5.57059668384447078051761608503, 6.61642633441396966534873934476, 6.99549741217795448239305223620, 7.74965213848326785022799319923, 8.911241264865647419466454365586
