| L(s) = 1 | − 1.90·3-s + 5-s + 1.15·7-s + 0.640·9-s + 4.77·11-s + 3.49·13-s − 1.90·15-s − 7.78·17-s + 1.91·19-s − 2.19·21-s − 6.30·23-s + 25-s + 4.50·27-s − 0.185·29-s − 9.77·31-s − 9.10·33-s + 1.15·35-s + 9.42·37-s − 6.66·39-s − 3.34·41-s − 1.84·43-s + 0.640·45-s − 1.21·47-s − 5.67·49-s + 14.8·51-s − 0.503·53-s + 4.77·55-s + ⋯ |
| L(s) = 1 | − 1.10·3-s + 0.447·5-s + 0.435·7-s + 0.213·9-s + 1.43·11-s + 0.968·13-s − 0.492·15-s − 1.88·17-s + 0.438·19-s − 0.479·21-s − 1.31·23-s + 0.200·25-s + 0.866·27-s − 0.0344·29-s − 1.75·31-s − 1.58·33-s + 0.194·35-s + 1.54·37-s − 1.06·39-s − 0.521·41-s − 0.281·43-s + 0.0955·45-s − 0.176·47-s − 0.810·49-s + 2.07·51-s − 0.0691·53-s + 0.643·55-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 | \( 1 + 1.90T + 3T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 0.185T + 29T^{2} \) |
| 31 | \( 1 + 9.77T + 31T^{2} \) |
| 37 | \( 1 - 9.42T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 0.503T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.60509814088554676604567789787, −6.73165951067572401589292787552, −6.12732423503704492009228398643, −5.89790281945093730876620716427, −4.78276934506531315006402302107, −4.28113230568115047551785113536, −3.36136209545697100809321416226, −2.01300154652783139825887234388, −1.32159077462744440224022369779, 0,
1.32159077462744440224022369779, 2.01300154652783139825887234388, 3.36136209545697100809321416226, 4.28113230568115047551785113536, 4.78276934506531315006402302107, 5.89790281945093730876620716427, 6.12732423503704492009228398643, 6.73165951067572401589292787552, 7.60509814088554676604567789787
