| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 6·11-s − 2·13-s + 4·14-s + 16-s − 17-s − 4·19-s − 6·22-s − 2·26-s + 4·28-s − 4·31-s + 32-s − 34-s + 4·37-s − 4·38-s − 6·41-s − 8·43-s − 6·44-s + 9·49-s − 2·52-s − 6·53-s + 4·56-s − 4·61-s − 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 1.80·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 1.27·22-s − 0.392·26-s + 0.755·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s + 0.657·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 0.904·44-s + 9/7·49-s − 0.277·52-s − 0.824·53-s + 0.534·56-s − 0.512·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.60748342359692475273842604773, −6.86213252269039360089208034740, −5.93429836379779240347032162521, −5.14122081735630950841586896220, −4.88704049759191627279528206271, −4.15060524565542301417202305165, −3.04463433857488825616535867090, −2.28700102226632678045675016123, −1.62721677829936424280869839867, 0,
1.62721677829936424280869839867, 2.28700102226632678045675016123, 3.04463433857488825616535867090, 4.15060524565542301417202305165, 4.88704049759191627279528206271, 5.14122081735630950841586896220, 5.93429836379779240347032162521, 6.86213252269039360089208034740, 7.60748342359692475273842604773
