| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 2·14-s + 15-s + 16-s + 2·17-s − 18-s − 20-s − 2·21-s + 7·23-s + 24-s + 25-s − 26-s − 27-s + 2·28-s − 30-s − 5·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s + 1.45·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.182·30-s − 0.898·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.98108686890310, −14.49991030444063, −13.83313348168071, −13.24731079921227, −12.56980907021752, −12.20576482928066, −11.56546897198026, −11.16782166220770, −10.84294671280729, −10.18938536343249, −9.743404353189130, −8.845275303591134, −8.704581889356128, −7.932875504816406, −7.504688791426854, −6.861324719779523, −6.524301434553527, −5.579207028885832, −5.138333893153738, −4.681023031015436, −3.642389893519770, −3.333132429244287, −2.263401279549576, −1.549165108279552, −0.9269797119630534, 0,
0.9269797119630534, 1.549165108279552, 2.263401279549576, 3.333132429244287, 3.642389893519770, 4.681023031015436, 5.138333893153738, 5.579207028885832, 6.524301434553527, 6.861324719779523, 7.504688791426854, 7.932875504816406, 8.704581889356128, 8.845275303591134, 9.743404353189130, 10.18938536343249, 10.84294671280729, 11.16782166220770, 11.56546897198026, 12.20576482928066, 12.56980907021752, 13.24731079921227, 13.83313348168071, 14.49991030444063, 14.98108686890310
