| L(s) = 1 | + 1.34·2-s − 0.185·4-s − 0.124·5-s − 7-s − 2.94·8-s − 0.168·10-s − 0.0232·11-s − 1.81·13-s − 1.34·14-s − 3.59·16-s − 5.85·17-s − 6.33·19-s + 0.0231·20-s − 0.0313·22-s − 5.01·23-s − 4.98·25-s − 2.44·26-s + 0.185·28-s + 5.62·29-s + 7.17·31-s + 1.04·32-s − 7.89·34-s + 0.124·35-s + 3.76·37-s − 8.53·38-s + 0.367·40-s + 1.39·41-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0926·4-s − 0.0558·5-s − 0.377·7-s − 1.04·8-s − 0.0531·10-s − 0.00702·11-s − 0.503·13-s − 0.360·14-s − 0.898·16-s − 1.42·17-s − 1.45·19-s + 0.00517·20-s − 0.00668·22-s − 1.04·23-s − 0.996·25-s − 0.479·26-s + 0.0350·28-s + 1.04·29-s + 1.28·31-s + 0.184·32-s − 1.35·34-s + 0.0211·35-s + 0.618·37-s − 1.38·38-s + 0.0581·40-s + 0.218·41-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
| \(L(1)\) |
\(\approx\) |
\(1.496431242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496431242\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 0.124T + 5T^{2} \) |
| 11 | \( 1 + 0.0232T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 + 6.33T + 19T^{2} \) |
| 23 | \( 1 + 5.01T + 23T^{2} \) |
| 29 | \( 1 - 5.62T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 1.62T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−7.892725997502043077921188554004, −6.75032684519108643259979830948, −6.41940733385052381435030355154, −5.73576991688648028161517371988, −4.88021087103376128697123147039, −4.17843780050006509403210129857, −3.90452041670535879199340391338, −2.62224233154957302046764960269, −2.25043515523329452513607377252, −0.49157246038772984982517942809,
0.49157246038772984982517942809, 2.25043515523329452513607377252, 2.62224233154957302046764960269, 3.90452041670535879199340391338, 4.17843780050006509403210129857, 4.88021087103376128697123147039, 5.73576991688648028161517371988, 6.41940733385052381435030355154, 6.75032684519108643259979830948, 7.892725997502043077921188554004
