| L(s) = 1 | − 3.27·3-s + 3.82·5-s − 1.10·7-s + 7.70·9-s + 4.05·11-s + 2.22·13-s − 12.5·15-s − 17-s − 5.58·19-s + 3.60·21-s − 8.79·23-s + 9.60·25-s − 15.3·27-s − 7.67·29-s + 4.75·31-s − 13.2·33-s − 4.20·35-s + 1.29·37-s − 7.27·39-s − 6.85·41-s − 2.65·43-s + 29.4·45-s − 11.7·47-s − 5.78·49-s + 3.27·51-s − 9.59·53-s + 15.4·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 1.70·5-s − 0.415·7-s + 2.56·9-s + 1.22·11-s + 0.617·13-s − 3.22·15-s − 0.242·17-s − 1.28·19-s + 0.785·21-s − 1.83·23-s + 1.92·25-s − 2.95·27-s − 1.42·29-s + 0.854·31-s − 2.30·33-s − 0.710·35-s + 0.213·37-s − 1.16·39-s − 1.07·41-s − 0.405·43-s + 4.38·45-s − 1.70·47-s − 0.826·49-s + 0.458·51-s − 1.31·53-s + 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 8.79T + 23T^{2} \) |
| 29 | \( 1 + 7.67T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 - 6.69T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 + 8.10T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−8.036853089570065392119885470304, −6.65177269654539609537899435459, −6.39992782595822110021552195882, −6.12420722710628541557637491767, −5.30253042040905605367410249107, −4.52390572870619788078411027963, −3.66422431408558999955487766107, −1.93279542512458449066619225962, −1.47039421325910029539278827958, 0,
1.47039421325910029539278827958, 1.93279542512458449066619225962, 3.66422431408558999955487766107, 4.52390572870619788078411027963, 5.30253042040905605367410249107, 6.12420722710628541557637491767, 6.39992782595822110021552195882, 6.65177269654539609537899435459, 8.036853089570065392119885470304
