L(s) = 1 | + 2.80·2-s + 5.88·4-s + 5-s + 7-s + 10.9·8-s + 2.80·10-s − 2.07·11-s − 5.69·13-s + 2.80·14-s + 18.8·16-s − 3.22·17-s − 3.29·19-s + 5.88·20-s − 5.83·22-s − 3.22·23-s + 25-s − 15.9·26-s + 5.88·28-s − 9.77·29-s − 0.396·31-s + 31.1·32-s − 9.04·34-s + 35-s + 7.61·37-s − 9.26·38-s + 10.9·40-s + 9.99·41-s + ⋯ |
L(s) = 1 | + 1.98·2-s + 2.94·4-s + 0.447·5-s + 0.377·7-s + 3.85·8-s + 0.888·10-s − 0.626·11-s − 1.57·13-s + 0.750·14-s + 4.72·16-s − 0.781·17-s − 0.756·19-s + 1.31·20-s − 1.24·22-s − 0.671·23-s + 0.200·25-s − 3.13·26-s + 1.11·28-s − 1.81·29-s − 0.0711·31-s + 5.51·32-s − 1.55·34-s + 0.169·35-s + 1.25·37-s − 1.50·38-s + 1.72·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.603053881\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.603053881\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 + 9.77T + 29T^{2} \) |
| 31 | \( 1 + 0.396T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 9.99T + 41T^{2} \) |
| 43 | \( 1 - 2.60T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 1.14T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 1.22T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 - 2.31T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.47998665761681332333329222647, −9.439668771069972414136643800895, −7.81769569059135292450059203127, −7.31692009769170341114840535216, −6.25457774171334357263383262780, −5.54590771120639111296309015692, −4.72753398411264665947075661909, −4.02868521570200853191611820911, −2.59086754541441002888504694443, −2.08449988678973411262231487890,
2.08449988678973411262231487890, 2.59086754541441002888504694443, 4.02868521570200853191611820911, 4.72753398411264665947075661909, 5.54590771120639111296309015692, 6.25457774171334357263383262780, 7.31692009769170341114840535216, 7.81769569059135292450059203127, 9.439668771069972414136643800895, 10.47998665761681332333329222647