| L(s) = 1 | − 0.695·3-s + 1.38·5-s − 7-s − 2.51·9-s + 0.420·11-s − 1.63·13-s − 0.962·15-s + 4.89·17-s + 3.98·19-s + 0.695·21-s + 6.81·23-s − 3.08·25-s + 3.83·27-s + 0.394·29-s + 5.18·31-s − 0.292·33-s − 1.38·35-s + 3.85·37-s + 1.13·39-s − 41-s + 4.97·43-s − 3.47·45-s − 3.46·47-s + 49-s − 3.40·51-s + 1.26·53-s + 0.581·55-s + ⋯ |
| L(s) = 1 | − 0.401·3-s + 0.618·5-s − 0.377·7-s − 0.838·9-s + 0.126·11-s − 0.452·13-s − 0.248·15-s + 1.18·17-s + 0.914·19-s + 0.151·21-s + 1.42·23-s − 0.617·25-s + 0.738·27-s + 0.0731·29-s + 0.931·31-s − 0.0509·33-s − 0.233·35-s + 0.633·37-s + 0.181·39-s − 0.156·41-s + 0.758·43-s − 0.518·45-s − 0.504·47-s + 0.142·49-s − 0.477·51-s + 0.173·53-s + 0.0784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457986029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457986029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.695T + 3T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 11 | \( 1 - 0.420T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 9.77T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 2.14T + 73T^{2} \) |
| 79 | \( 1 - 7.49T + 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 + 3.54T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.760919092880539190228717967533, −9.188476183999035449546017595509, −8.150074692988334777751503314544, −7.26500978666693746983047464532, −6.29460277889299611043740212879, −5.59229101756885778216735125792, −4.89370930545861141548776221433, −3.42711224397772127544514351172, −2.56764665288137933428110888714, −0.952968701710187202164051475581,
0.952968701710187202164051475581, 2.56764665288137933428110888714, 3.42711224397772127544514351172, 4.89370930545861141548776221433, 5.59229101756885778216735125792, 6.29460277889299611043740212879, 7.26500978666693746983047464532, 8.150074692988334777751503314544, 9.188476183999035449546017595509, 9.760919092880539190228717967533
