| L(s) = 1 | + 3-s − 5-s + 1.37·7-s + 9-s − 1.37·11-s − 13-s − 15-s − 5.37·17-s − 2·19-s + 1.37·21-s + 3.37·23-s + 25-s + 27-s + 8.74·29-s − 4.74·31-s − 1.37·33-s − 1.37·35-s − 1.37·37-s − 39-s + 5.37·41-s − 6.74·43-s − 45-s − 8.74·47-s − 5.11·49-s − 5.37·51-s − 13.3·53-s + 1.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.518·7-s + 0.333·9-s − 0.413·11-s − 0.277·13-s − 0.258·15-s − 1.30·17-s − 0.458·19-s + 0.299·21-s + 0.703·23-s + 0.200·25-s + 0.192·27-s + 1.62·29-s − 0.852·31-s − 0.238·33-s − 0.231·35-s − 0.225·37-s − 0.160·39-s + 0.839·41-s − 1.02·43-s − 0.149·45-s − 1.27·47-s − 0.730·49-s − 0.752·51-s − 1.83·53-s + 0.185·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
| 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
−7.944402474334000709144354697254, −6.87614989520827523125175693594, −6.63782722916762836552082879154, −5.34780925261638376154217830821, −4.71289872935930689991358443695, −4.10707421330271105895252393809, −3.10888672306195319454915802463, −2.40479157068419456330073465136, −1.44043133065990194696224928943, 0,
1.44043133065990194696224928943, 2.40479157068419456330073465136, 3.10888672306195319454915802463, 4.10707421330271105895252393809, 4.71289872935930689991358443695, 5.34780925261638376154217830821, 6.63782722916762836552082879154, 6.87614989520827523125175693594, 7.944402474334000709144354697254
