| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 3·14-s − 15-s + 16-s − 17-s + 18-s − 5·19-s + 20-s − 3·21-s + 22-s + 7·23-s − 24-s + 25-s + 26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.640973812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.640973812\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.08072534316222640609880179955, −8.960320790585387478147860975644, −8.166076802972896131660435360600, −7.06424021617212550510788810925, −6.39054737840711598284307932890, −5.42145831975423179442916360935, −4.78434526310804408668386291359, −3.89424117562758728523112409100, −2.42537307729511615919253401926, −1.31389015134512166909255491495,
1.31389015134512166909255491495, 2.42537307729511615919253401926, 3.89424117562758728523112409100, 4.78434526310804408668386291359, 5.42145831975423179442916360935, 6.39054737840711598284307932890, 7.06424021617212550510788810925, 8.166076802972896131660435360600, 8.960320790585387478147860975644, 10.08072534316222640609880179955
