L(s) = 1 | + 2-s + 3-s + 4-s − 1.31·5-s + 6-s + 3.43·7-s + 8-s + 9-s − 1.31·10-s − 3.11·11-s + 12-s + 13-s + 3.43·14-s − 1.31·15-s + 16-s + 6.00·17-s + 18-s + 2.43·19-s − 1.31·20-s + 3.43·21-s − 3.11·22-s + 5.59·23-s + 24-s − 3.27·25-s + 26-s + 27-s + 3.43·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.588·5-s + 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s − 0.415·10-s − 0.938·11-s + 0.288·12-s + 0.277·13-s + 0.918·14-s − 0.339·15-s + 0.250·16-s + 1.45·17-s + 0.235·18-s + 0.559·19-s − 0.294·20-s + 0.749·21-s − 0.663·22-s + 1.16·23-s + 0.204·24-s − 0.654·25-s + 0.196·26-s + 0.192·27-s + 0.649·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.745370473\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.745370473\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 5.59T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 - 0.544T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 5.36T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 9.93T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 8.21T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 - 9.40T + 97T^{2} \) |
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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
−7.73205348189581680761095630888, −7.42838976089545988240800589107, −6.45680447199628863059825921547, −5.38515031112244151712137922853, −5.07705746320266594520154277542, −4.32288386591858959247280443468, −3.43159642423204129549326558962, −2.92414121227602567091922297112, −1.88384989744958886768245026672, −1.00269298909760475608504662762,
1.00269298909760475608504662762, 1.88384989744958886768245026672, 2.92414121227602567091922297112, 3.43159642423204129549326558962, 4.32288386591858959247280443468, 5.07705746320266594520154277542, 5.38515031112244151712137922853, 6.45680447199628863059825921547, 7.42838976089545988240800589107, 7.73205348189581680761095630888