| L(s) = 1 | − 1.14·2-s − 0.686·4-s − 0.461·5-s − 3.03·7-s + 3.07·8-s + 0.528·10-s − 1.24·11-s + 3.02·13-s + 3.47·14-s − 2.15·16-s − 1.56·17-s − 0.758·19-s + 0.316·20-s + 1.42·22-s − 23-s − 4.78·25-s − 3.47·26-s + 2.08·28-s − 29-s + 5.58·31-s − 3.68·32-s + 1.79·34-s + 1.39·35-s + 11.3·37-s + 0.869·38-s − 1.42·40-s − 4.22·41-s + ⋯ |
| L(s) = 1 | − 0.810·2-s − 0.343·4-s − 0.206·5-s − 1.14·7-s + 1.08·8-s + 0.167·10-s − 0.373·11-s + 0.839·13-s + 0.928·14-s − 0.539·16-s − 0.379·17-s − 0.174·19-s + 0.0707·20-s + 0.303·22-s − 0.208·23-s − 0.957·25-s − 0.680·26-s + 0.393·28-s − 0.185·29-s + 1.00·31-s − 0.651·32-s + 0.307·34-s + 0.236·35-s + 1.87·37-s + 0.141·38-s − 0.224·40-s − 0.659·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 5 | \( 1 + 0.461T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 0.758T + 19T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 3.44T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.73797642166803545091649396841, −7.30005532296840427709642244855, −6.24628146160713662711227142045, −5.86622756322773088128940692720, −4.66241421069803401228728911981, −4.05551646897873037495244478953, −3.23729207210556060952426666471, −2.21731160953064931084625872091, −0.982056858634458636150974985228, 0,
0.982056858634458636150974985228, 2.21731160953064931084625872091, 3.23729207210556060952426666471, 4.05551646897873037495244478953, 4.66241421069803401228728911981, 5.86622756322773088128940692720, 6.24628146160713662711227142045, 7.30005532296840427709642244855, 7.73797642166803545091649396841
