| L(s) = 1 | + 15.3·5-s + 7·7-s + 7.38·11-s − 67.5·13-s − 138.·17-s + 116.·19-s − 133.·23-s + 111.·25-s − 120·29-s − 295.·31-s + 107.·35-s − 337.·37-s − 409.·41-s − 346.·43-s + 345.·47-s + 49·49-s + 381.·53-s + 113.·55-s + 438.·59-s + 707.·61-s − 1.03e3·65-s − 264.·67-s + 391.·71-s − 237.·73-s + 51.6·77-s + 1.14e3·79-s − 49.1·83-s + ⋯ |
| L(s) = 1 | + 1.37·5-s + 0.377·7-s + 0.202·11-s − 1.44·13-s − 1.97·17-s + 1.40·19-s − 1.21·23-s + 0.892·25-s − 0.768·29-s − 1.71·31-s + 0.519·35-s − 1.50·37-s − 1.55·41-s − 1.22·43-s + 1.07·47-s + 0.142·49-s + 0.990·53-s + 0.278·55-s + 0.966·59-s + 1.48·61-s − 1.98·65-s − 0.483·67-s + 0.655·71-s − 0.380·73-s + 0.0764·77-s + 1.62·79-s − 0.0649·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 15.3T + 125T^{2} \) |
| 11 | \( 1 - 7.38T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 138.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 133.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 120T + 2.43e4T^{2} \) |
| 31 | \( 1 + 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 381.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 707.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 391.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 49.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 397.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.319319661218395520632491079799, −8.520217451217267864740402314161, −7.27447047465421077554938594486, −6.71949442209246230618945418593, −5.49473398606423480705921867425, −5.09178035479241507383650380110, −3.78483314646981575959832555112, −2.29535447797666443774611680904, −1.80381636850921785134360816542, 0,
1.80381636850921785134360816542, 2.29535447797666443774611680904, 3.78483314646981575959832555112, 5.09178035479241507383650380110, 5.49473398606423480705921867425, 6.71949442209246230618945418593, 7.27447047465421077554938594486, 8.520217451217267864740402314161, 9.319319661218395520632491079799
