L(s) = 1 | + 2.40·2-s + 3.77·4-s + 0.666·5-s + 7-s + 4.25·8-s + 1.60·10-s + 5.05·11-s + 5.76·13-s + 2.40·14-s + 2.68·16-s + 0.396·17-s + 3.56·19-s + 2.51·20-s + 12.1·22-s + 4.24·23-s − 4.55·25-s + 13.8·26-s + 3.77·28-s − 0.519·29-s + 3.84·31-s − 2.06·32-s + 0.952·34-s + 0.666·35-s − 7.00·37-s + 8.57·38-s + 2.83·40-s − 3.06·41-s + ⋯ |
L(s) = 1 | + 1.69·2-s + 1.88·4-s + 0.298·5-s + 0.377·7-s + 1.50·8-s + 0.506·10-s + 1.52·11-s + 1.60·13-s + 0.642·14-s + 0.670·16-s + 0.0961·17-s + 0.818·19-s + 0.562·20-s + 2.58·22-s + 0.884·23-s − 0.911·25-s + 2.71·26-s + 0.712·28-s − 0.0965·29-s + 0.690·31-s − 0.365·32-s + 0.163·34-s + 0.112·35-s − 1.15·37-s + 1.39·38-s + 0.448·40-s − 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.214979944\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.214979944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 5 | \( 1 - 0.666T + 5T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 - 0.396T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 0.519T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + 7.00T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 6.50T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−7.56965758400123316826975253666, −6.65678658190042488559105347950, −6.39886004104044680890076803313, −5.65894922387849866611508512410, −5.01150169070710782104547802994, −4.26955642297587379507685694949, −3.54034856902392427247847265423, −3.17371158144119946530302661912, −1.83807406616482213668882010054, −1.27817136371281216808926895726,
1.27817136371281216808926895726, 1.83807406616482213668882010054, 3.17371158144119946530302661912, 3.54034856902392427247847265423, 4.26955642297587379507685694949, 5.01150169070710782104547802994, 5.65894922387849866611508512410, 6.39886004104044680890076803313, 6.65678658190042488559105347950, 7.56965758400123316826975253666