| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 4·11-s + 2·13-s − 4·14-s + 16-s − 2·17-s − 19-s + 4·22-s − 8·23-s + 2·26-s − 4·28-s − 6·29-s + 4·31-s + 32-s − 2·34-s + 10·37-s − 38-s + 2·41-s − 12·43-s + 4·44-s − 8·46-s + 9·49-s + 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.852·22-s − 1.66·23-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1.64·37-s − 0.162·38-s + 0.312·41-s − 1.82·43-s + 0.603·44-s − 1.17·46-s + 9/7·49-s + 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.15770912822291761947965035369, −6.48801873778727791621256291858, −6.19563343316073825191709338723, −5.55305678482459290140237738191, −4.29915055845805882478815828671, −3.96564951081572590139725777543, −3.24728229749007396311542791168, −2.40176698563740779955778170375, −1.38369596314356637117144031912, 0,
1.38369596314356637117144031912, 2.40176698563740779955778170375, 3.24728229749007396311542791168, 3.96564951081572590139725777543, 4.29915055845805882478815828671, 5.55305678482459290140237738191, 6.19563343316073825191709338723, 6.48801873778727791621256291858, 7.15770912822291761947965035369
