| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s + 2·13-s + 14-s + 2·15-s + 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s + 21-s − 23-s + 24-s − 25-s − 2·26-s − 27-s − 28-s + 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9981934612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9981934612\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.36794530112170, −12.99340586504185, −12.49085892327057, −12.02928899028395, −11.65707005453137, −10.96824927020827, −10.89815921427642, −10.12455293919369, −9.768503156139404, −9.238042734322900, −8.583898451744441, −8.150108500857149, −7.630914202747838, −7.288611313874080, −6.613185461100911, −6.105712341338618, −5.663969007930634, −5.063952626445933, −4.212890488433383, −3.801520055947894, −3.318705086857256, −2.492187904031475, −1.793028537005649, −0.9337319952227127, −0.4543487734814837,
0.4543487734814837, 0.9337319952227127, 1.793028537005649, 2.492187904031475, 3.318705086857256, 3.801520055947894, 4.212890488433383, 5.063952626445933, 5.663969007930634, 6.105712341338618, 6.613185461100911, 7.288611313874080, 7.630914202747838, 8.150108500857149, 8.583898451744441, 9.238042734322900, 9.768503156139404, 10.12455293919369, 10.89815921427642, 10.96824927020827, 11.65707005453137, 12.02928899028395, 12.49085892327057, 12.99340586504185, 13.36794530112170
