| L(s) = 1 | + 1.28·3-s + 4.24·5-s + 3.37·7-s − 1.34·9-s − 0.495·11-s + 3.53·13-s + 5.46·15-s − 3.96·17-s + 4.34·21-s + 0.943·23-s + 13.0·25-s − 5.59·27-s + 2.66·29-s − 6.83·31-s − 0.638·33-s + 14.3·35-s − 6.73·37-s + 4.54·39-s + 2.42·41-s + 11.9·43-s − 5.69·45-s + 1.11·47-s + 4.37·49-s − 5.10·51-s − 1.01·53-s − 2.10·55-s − 6.59·59-s + ⋯ |
| L(s) = 1 | + 0.743·3-s + 1.89·5-s + 1.27·7-s − 0.447·9-s − 0.149·11-s + 0.979·13-s + 1.41·15-s − 0.962·17-s + 0.947·21-s + 0.196·23-s + 2.60·25-s − 1.07·27-s + 0.495·29-s − 1.22·31-s − 0.111·33-s + 2.41·35-s − 1.10·37-s + 0.728·39-s + 0.378·41-s + 1.82·43-s − 0.849·45-s + 0.163·47-s + 0.624·49-s − 0.715·51-s − 0.139·53-s − 0.283·55-s − 0.858·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.894530273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894530273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 0.495T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 23 | \( 1 - 0.943T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 9.42T + 71T^{2} \) |
| 73 | \( 1 - 3.25T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 1.43T + 97T^{2} \) |
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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
−8.855327640062752096923213442015, −8.299927900915131473329524989977, −7.30546095766076837721566038966, −6.36877480461549265663117610835, −5.64260446322133116155292398913, −5.09493570849520341007228536818, −4.00918749203631431503863482452, −2.77185463836844962906836354901, −2.09689673643395765743177732814, −1.36353196031020419153457384015,
1.36353196031020419153457384015, 2.09689673643395765743177732814, 2.77185463836844962906836354901, 4.00918749203631431503863482452, 5.09493570849520341007228536818, 5.64260446322133116155292398913, 6.36877480461549265663117610835, 7.30546095766076837721566038966, 8.299927900915131473329524989977, 8.855327640062752096923213442015
