| L(s) = 1 | + 2.23·3-s + 0.381·7-s + 2.00·9-s + 3.23·11-s − 5.61·13-s − 0.763·17-s − 5.38·19-s + 0.854·21-s − 3.47·23-s − 2.23·27-s + 6.61·29-s − 8.70·31-s + 7.23·33-s + 3.76·37-s − 12.5·39-s − 7.70·41-s + 6.61·43-s + 6.85·47-s − 6.85·49-s − 1.70·51-s + 9.94·53-s − 12.0·57-s − 4.61·59-s + 9.18·61-s + 0.763·63-s + 0.291·67-s − 7.76·69-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 0.144·7-s + 0.666·9-s + 0.975·11-s − 1.55·13-s − 0.185·17-s − 1.23·19-s + 0.186·21-s − 0.723·23-s − 0.430·27-s + 1.22·29-s − 1.56·31-s + 1.25·33-s + 0.618·37-s − 2.01·39-s − 1.20·41-s + 1.00·43-s + 0.999·47-s − 0.979·49-s − 0.239·51-s + 1.36·53-s − 1.59·57-s − 0.601·59-s + 1.17·61-s + 0.0962·63-s + 0.0356·67-s − 0.934·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 47 | \( 1 - 6.85T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 - 0.291T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
| 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
−7.31350791882061690839223751459, −6.90114258173941220337504870294, −6.02575128026533031088642919863, −5.17468040160288130637742916951, −4.20714383187973290186043214489, −3.93329522069517723229965956271, −2.80759311475309955014554407911, −2.33566362265028674344458753844, −1.53052124860356232780969085070, 0,
1.53052124860356232780969085070, 2.33566362265028674344458753844, 2.80759311475309955014554407911, 3.93329522069517723229965956271, 4.20714383187973290186043214489, 5.17468040160288130637742916951, 6.02575128026533031088642919863, 6.90114258173941220337504870294, 7.31350791882061690839223751459
