L(s) = 1 | − 2-s − 0.296·3-s + 4-s − 5-s + 0.296·6-s − 4.13·7-s − 8-s − 2.91·9-s + 10-s − 0.822·11-s − 0.296·12-s + 3.99·13-s + 4.13·14-s + 0.296·15-s + 16-s + 0.706·17-s + 2.91·18-s − 7.10·19-s − 20-s + 1.22·21-s + 0.822·22-s + 1.64·23-s + 0.296·24-s + 25-s − 3.99·26-s + 1.75·27-s − 4.13·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.170·3-s + 0.5·4-s − 0.447·5-s + 0.120·6-s − 1.56·7-s − 0.353·8-s − 0.970·9-s + 0.316·10-s − 0.248·11-s − 0.0854·12-s + 1.10·13-s + 1.10·14-s + 0.0764·15-s + 0.250·16-s + 0.171·17-s + 0.686·18-s − 1.62·19-s − 0.223·20-s + 0.267·21-s + 0.175·22-s + 0.344·23-s + 0.0604·24-s + 0.200·25-s − 0.784·26-s + 0.336·27-s − 0.781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 0.296T + 3T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 0.822T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 0.706T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 + 9.56T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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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
−7.974488665400308190635511971989, −6.87806021181235108584152041354, −6.20217901327141031579504922039, −6.14431444483972347727343247815, −4.83027968520415133899413166491, −3.85058408881772654402889762758, −3.06197689106644775263570533539, −2.49314727871740895920066853797, −0.935200654915718549818550946309, 0,
0.935200654915718549818550946309, 2.49314727871740895920066853797, 3.06197689106644775263570533539, 3.85058408881772654402889762758, 4.83027968520415133899413166491, 6.14431444483972347727343247815, 6.20217901327141031579504922039, 6.87806021181235108584152041354, 7.974488665400308190635511971989