| L(s) = 1 | − 2.41·3-s + 5-s − 2.82·7-s + 2.82·9-s + 0.414·11-s + 3.82·13-s − 2.41·15-s + 0.828·17-s − 6·19-s + 6.82·21-s + 3.65·23-s − 4·25-s + 0.414·27-s − 29-s + 10.0·31-s − 0.999·33-s − 2.82·35-s + 4·37-s − 9.24·39-s − 4.48·41-s − 3.58·43-s + 2.82·45-s − 3.24·47-s + 1.00·49-s − 1.99·51-s − 9.48·53-s + 0.414·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s − 1.06·7-s + 0.942·9-s + 0.124·11-s + 1.06·13-s − 0.623·15-s + 0.200·17-s − 1.37·19-s + 1.49·21-s + 0.762·23-s − 0.800·25-s + 0.0797·27-s − 0.185·29-s + 1.80·31-s − 0.174·33-s − 0.478·35-s + 0.657·37-s − 1.48·39-s − 0.700·41-s − 0.546·43-s + 0.421·45-s − 0.472·47-s + 0.142·49-s − 0.280·51-s − 1.30·53-s + 0.0558·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−8.935108489628942822524308248902, −8.109341981290463959091062380491, −6.76966144069386946384857676520, −6.35496258510786618587539178043, −5.88224680086553034473616615342, −4.90316089408612750083044511199, −3.96437482878671976976643338134, −2.83904810850298681045805294085, −1.33705204080188680039538287194, 0,
1.33705204080188680039538287194, 2.83904810850298681045805294085, 3.96437482878671976976643338134, 4.90316089408612750083044511199, 5.88224680086553034473616615342, 6.35496258510786618587539178043, 6.76966144069386946384857676520, 8.109341981290463959091062380491, 8.935108489628942822524308248902
