| L(s) = 1 | − 2.03·2-s + 2.19·3-s + 2.16·4-s − 5-s − 4.48·6-s − 0.326·8-s + 1.83·9-s + 2.03·10-s − 11-s + 4.75·12-s + 5.64·13-s − 2.19·15-s − 3.65·16-s + 4.19·17-s − 3.75·18-s − 7.52·19-s − 2.16·20-s + 2.03·22-s + 7.79·23-s − 0.719·24-s + 25-s − 11.5·26-s − 2.55·27-s + 0.894·29-s + 4.48·30-s + 5.11·31-s + 8.10·32-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.27·3-s + 1.08·4-s − 0.447·5-s − 1.83·6-s − 0.115·8-s + 0.613·9-s + 0.645·10-s − 0.301·11-s + 1.37·12-s + 1.56·13-s − 0.568·15-s − 0.913·16-s + 1.01·17-s − 0.884·18-s − 1.72·19-s − 0.483·20-s + 0.434·22-s + 1.62·23-s − 0.146·24-s + 0.200·25-s − 2.25·26-s − 0.491·27-s + 0.166·29-s + 0.819·30-s + 0.919·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.303040568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303040568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 - 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.894T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 0.287T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
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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
−8.596139297195950556021570047902, −8.425121509201030551853496042141, −7.75140928429590835603456777257, −6.95599271332363948083128516098, −6.12056207879903811878368406223, −4.74843488477857179583769745983, −3.73607902564757029741014630834, −2.96012581075396018881275332609, −1.91965240162067563969570841061, −0.859915714721390347854801332410,
0.859915714721390347854801332410, 1.91965240162067563969570841061, 2.96012581075396018881275332609, 3.73607902564757029741014630834, 4.74843488477857179583769745983, 6.12056207879903811878368406223, 6.95599271332363948083128516098, 7.75140928429590835603456777257, 8.425121509201030551853496042141, 8.596139297195950556021570047902
