L(s) = 1 | − 2-s − 3-s + 4-s − 1.65·5-s + 6-s + 0.184·7-s − 8-s + 9-s + 1.65·10-s − 4.34·11-s − 12-s + 6.47·13-s − 0.184·14-s + 1.65·15-s + 16-s − 2.12·17-s − 18-s − 1.65·20-s − 0.184·21-s + 4.34·22-s + 0.106·23-s + 24-s − 2.26·25-s − 6.47·26-s − 27-s + 0.184·28-s − 3.98·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.739·5-s + 0.408·6-s + 0.0698·7-s − 0.353·8-s + 0.333·9-s + 0.522·10-s − 1.31·11-s − 0.288·12-s + 1.79·13-s − 0.0493·14-s + 0.426·15-s + 0.250·16-s − 0.514·17-s − 0.235·18-s − 0.369·20-s − 0.0403·21-s + 0.926·22-s + 0.0221·23-s + 0.204·24-s − 0.453·25-s − 1.26·26-s − 0.192·27-s + 0.0349·28-s − 0.740·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6811236354\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6811236354\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 0.184T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 23 | \( 1 - 0.106T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 - 3.22T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 - 0.0418T + 43T^{2} \) |
| 47 | \( 1 + 7.92T + 47T^{2} \) |
| 53 | \( 1 - 9.21T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 - 4.04T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.974536826705302242679938269273, −8.153536459252923568363488940050, −7.78174084811435678533601274918, −6.77893836226605670961602814473, −6.04101984839712838059217422723, −5.21985410411543365466991427871, −4.13605496995761638505157952239, −3.25493163809521606105904737843, −1.94643732661164759335011404428, −0.60975817559061267230126568859,
0.60975817559061267230126568859, 1.94643732661164759335011404428, 3.25493163809521606105904737843, 4.13605496995761638505157952239, 5.21985410411543365466991427871, 6.04101984839712838059217422723, 6.77893836226605670961602814473, 7.78174084811435678533601274918, 8.153536459252923568363488940050, 8.974536826705302242679938269273