L(s) = 1 | − 0.507·2-s + 7.53·3-s − 7.74·4-s − 5·5-s − 3.82·6-s − 10.8·7-s + 7.98·8-s + 29.8·9-s + 2.53·10-s + 11·11-s − 58.3·12-s − 1.60·13-s + 5.48·14-s − 37.6·15-s + 57.8·16-s + 71.2·17-s − 15.1·18-s − 19·19-s + 38.7·20-s − 81.5·21-s − 5.58·22-s − 83.9·23-s + 60.2·24-s + 25·25-s + 0.813·26-s + 21.4·27-s + 83.7·28-s + ⋯ |
L(s) = 1 | − 0.179·2-s + 1.45·3-s − 0.967·4-s − 0.447·5-s − 0.260·6-s − 0.584·7-s + 0.353·8-s + 1.10·9-s + 0.0802·10-s + 0.301·11-s − 1.40·12-s − 0.0341·13-s + 0.104·14-s − 0.648·15-s + 0.904·16-s + 1.01·17-s − 0.198·18-s − 0.229·19-s + 0.432·20-s − 0.847·21-s − 0.0540·22-s − 0.761·23-s + 0.512·24-s + 0.200·25-s + 0.00613·26-s + 0.152·27-s + 0.565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 + 0.507T + 8T^{2} \) |
| 3 | \( 1 - 7.53T + 27T^{2} \) |
| 7 | \( 1 + 10.8T + 343T^{2} \) |
| 13 | \( 1 + 1.60T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 83.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 80.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 440.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 29.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 209.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 525.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 895.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.66e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 420.T + 9.12e5T^{2} \) |
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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
−9.063327855423028260182298443158, −8.273722861227687219974318692044, −7.906707930557353134801962713229, −6.85305892905844222184707233444, −5.60395004950110671548361586693, −4.37800653394707491692500549996, −3.64566635955975776387786631806, −2.92443167528093496857004550897, −1.47027909843892168386414759782, 0,
1.47027909843892168386414759782, 2.92443167528093496857004550897, 3.64566635955975776387786631806, 4.37800653394707491692500549996, 5.60395004950110671548361586693, 6.85305892905844222184707233444, 7.906707930557353134801962713229, 8.273722861227687219974318692044, 9.063327855423028260182298443158