| L(s) = 1 | + 2-s − 3-s + 4-s − 3.89·5-s − 6-s − 4.93·7-s + 8-s + 9-s − 3.89·10-s + 6.46·11-s − 12-s + 1.90·13-s − 4.93·14-s + 3.89·15-s + 16-s − 4.76·17-s + 18-s + 19-s − 3.89·20-s + 4.93·21-s + 6.46·22-s − 4.82·23-s − 24-s + 10.1·25-s + 1.90·26-s − 27-s − 4.93·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.74·5-s − 0.408·6-s − 1.86·7-s + 0.353·8-s + 0.333·9-s − 1.23·10-s + 1.94·11-s − 0.288·12-s + 0.527·13-s − 1.31·14-s + 1.00·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s + 0.229·19-s − 0.870·20-s + 1.07·21-s + 1.37·22-s − 1.00·23-s − 0.204·24-s + 2.03·25-s + 0.372·26-s − 0.192·27-s − 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 4.93T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 + 0.937T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 - 0.539T + 47T^{2} \) |
| 59 | \( 1 - 8.13T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
| 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
−7.37548259129875440561166370693, −6.78490642051138027087208926571, −6.42478692317612227757893716284, −5.77837323957269945980966174396, −4.41572959907570953093526347237, −4.00906761184620055503057890246, −3.62619783924058391463477026927, −2.69326641427338254616445625241, −1.07260975283014983490707644190, 0,
1.07260975283014983490707644190, 2.69326641427338254616445625241, 3.62619783924058391463477026927, 4.00906761184620055503057890246, 4.41572959907570953093526347237, 5.77837323957269945980966174396, 6.42478692317612227757893716284, 6.78490642051138027087208926571, 7.37548259129875440561166370693
