L(s) = 1 | − 2.05·2-s − 3-s + 2.23·4-s − 1.05·5-s + 2.05·6-s + 7-s − 0.492·8-s + 9-s + 2.18·10-s − 2.23·12-s − 3.80·13-s − 2.05·14-s + 1.05·15-s − 3.46·16-s + 2.56·17-s − 2.05·18-s + 0.180·19-s − 2.37·20-s − 21-s + 0.433·23-s + 0.492·24-s − 3.87·25-s + 7.83·26-s − 27-s + 2.23·28-s + 9.03·29-s − 2.18·30-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.11·4-s − 0.473·5-s + 0.840·6-s + 0.377·7-s − 0.174·8-s + 0.333·9-s + 0.689·10-s − 0.646·12-s − 1.05·13-s − 0.550·14-s + 0.273·15-s − 0.866·16-s + 0.622·17-s − 0.485·18-s + 0.0413·19-s − 0.530·20-s − 0.218·21-s + 0.0904·23-s + 0.100·24-s − 0.775·25-s + 1.53·26-s − 0.192·27-s + 0.423·28-s + 1.67·29-s − 0.398·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4878561775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4878561775\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.180T + 19T^{2} \) |
| 23 | \( 1 - 0.433T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 0.492T + 31T^{2} \) |
| 37 | \( 1 + 0.775T + 37T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 - 0.502T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 + 1.37T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.906484376052808385367878371926, −8.086619693786289585833277201737, −7.58126257445685580076868216247, −6.95539795114499794102712592138, −6.00064150197103967236295116082, −4.94678474437504982025975567405, −4.26705997544784764621437540156, −2.85287433775151158513803708994, −1.68605987517153223799192284958, −0.57794194531910888766695245795,
0.57794194531910888766695245795, 1.68605987517153223799192284958, 2.85287433775151158513803708994, 4.26705997544784764621437540156, 4.94678474437504982025975567405, 6.00064150197103967236295116082, 6.95539795114499794102712592138, 7.58126257445685580076868216247, 8.086619693786289585833277201737, 8.906484376052808385367878371926