| L(s) = 1 | + 2-s − 3-s + 4-s + 3.94·5-s − 6-s + 7-s + 8-s + 9-s + 3.94·10-s − 2.25·11-s − 12-s − 6.54·13-s + 14-s − 3.94·15-s + 16-s + 0.785·17-s + 18-s − 7.36·19-s + 3.94·20-s − 21-s − 2.25·22-s − 0.429·23-s − 24-s + 10.5·25-s − 6.54·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.76·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.24·10-s − 0.680·11-s − 0.288·12-s − 1.81·13-s + 0.267·14-s − 1.01·15-s + 0.250·16-s + 0.190·17-s + 0.235·18-s − 1.68·19-s + 0.882·20-s − 0.218·21-s − 0.481·22-s − 0.0896·23-s − 0.204·24-s + 2.11·25-s − 1.28·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 3.94T + 5T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 13 | \( 1 + 6.54T + 13T^{2} \) |
| 17 | \( 1 - 0.785T + 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 0.429T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 - 6.37T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 + 5.22T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.31766745208172667204199147244, −6.57046892345318535832458309168, −5.90325158096652512232666395446, −5.41510820991892019535648301601, −4.89685806730296848803133574872, −4.19620470666328109819635298490, −2.88934261894167904424379755243, −2.13281479662611483238953311693, −1.73093622779119120691297691824, 0,
1.73093622779119120691297691824, 2.13281479662611483238953311693, 2.88934261894167904424379755243, 4.19620470666328109819635298490, 4.89685806730296848803133574872, 5.41510820991892019535648301601, 5.90325158096652512232666395446, 6.57046892345318535832458309168, 7.31766745208172667204199147244
