L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 2·13-s − 14-s + 16-s + 17-s − 2·20-s − 3·23-s − 25-s + 2·26-s − 28-s + 29-s − 8·31-s + 32-s + 34-s + 2·35-s + 37-s − 2·40-s + 11·41-s − 43-s − 3·46-s + 5·47-s − 6·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.447·20-s − 0.625·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.185·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 0.164·37-s − 0.316·40-s + 1.71·41-s − 0.152·43-s − 0.442·46-s + 0.729·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.58609577752968387421247847814, −6.98176517625217315773370909160, −6.06709492941274069707005692743, −5.64626457175549455940158970244, −4.59079443723511108256892397017, −3.97077412003693029121413392730, −3.41264209032169288900189102891, −2.52857371649904627619293970964, −1.38696560534371501963944497289, 0,
1.38696560534371501963944497289, 2.52857371649904627619293970964, 3.41264209032169288900189102891, 3.97077412003693029121413392730, 4.59079443723511108256892397017, 5.64626457175549455940158970244, 6.06709492941274069707005692743, 6.98176517625217315773370909160, 7.58609577752968387421247847814