| L(s) = 1 | − 0.429·3-s − 5-s − 1.03·7-s − 2.81·9-s + 2.79·11-s + 4.59·13-s + 0.429·15-s − 2.11·17-s − 3.83·19-s + 0.442·21-s − 4.89·23-s + 25-s + 2.49·27-s + 0.574·29-s + 9.21·31-s − 1.20·33-s + 1.03·35-s − 4.56·37-s − 1.97·39-s + 9.36·41-s + 7.45·43-s + 2.81·45-s − 4.05·47-s − 5.93·49-s + 0.905·51-s − 5.14·53-s − 2.79·55-s + ⋯ |
| L(s) = 1 | − 0.247·3-s − 0.447·5-s − 0.389·7-s − 0.938·9-s + 0.843·11-s + 1.27·13-s + 0.110·15-s − 0.511·17-s − 0.879·19-s + 0.0965·21-s − 1.02·23-s + 0.200·25-s + 0.480·27-s + 0.106·29-s + 1.65·31-s − 0.208·33-s + 0.174·35-s − 0.750·37-s − 0.315·39-s + 1.46·41-s + 1.13·43-s + 0.419·45-s − 0.590·47-s − 0.848·49-s + 0.126·51-s − 0.706·53-s − 0.377·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 0.429T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 0.574T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 - 7.45T + 43T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 0.885T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 5.95T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 3.06T + 83T^{2} \) |
| 89 | \( 1 + 4.05T + 89T^{2} \) |
| 97 | \( 1 - 7.56T + 97T^{2} \) |
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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
−7.977082208133832698863941803131, −6.74222313530587243648662885510, −6.29082266074904255605913770819, −5.85754578577231465160893247733, −4.68236702471204141632711952638, −4.04198696534628248894685979693, −3.31760047689634245401203207082, −2.39632796486481338664602200516, −1.18153605182581714075511538337, 0,
1.18153605182581714075511538337, 2.39632796486481338664602200516, 3.31760047689634245401203207082, 4.04198696534628248894685979693, 4.68236702471204141632711952638, 5.85754578577231465160893247733, 6.29082266074904255605913770819, 6.74222313530587243648662885510, 7.977082208133832698863941803131
