| L(s) = 1 | − 1.27·3-s + 2.87·5-s − 7-s − 1.38·9-s − 1.27·11-s + 13-s − 3.65·15-s + 3.10·17-s + 0.870·19-s + 1.27·21-s + 0.402·23-s + 3.23·25-s + 5.57·27-s + 6.52·29-s + 5.79·31-s + 1.61·33-s − 2.87·35-s + 0.727·37-s − 1.27·39-s − 6.12·41-s + 0.237·43-s − 3.96·45-s + 5.79·47-s + 49-s − 3.95·51-s − 2.52·53-s − 3.65·55-s + ⋯ |
| L(s) = 1 | − 0.734·3-s + 1.28·5-s − 0.377·7-s − 0.460·9-s − 0.383·11-s + 0.277·13-s − 0.943·15-s + 0.753·17-s + 0.199·19-s + 0.277·21-s + 0.0839·23-s + 0.647·25-s + 1.07·27-s + 1.21·29-s + 1.04·31-s + 0.281·33-s − 0.485·35-s + 0.119·37-s − 0.203·39-s − 0.955·41-s + 0.0361·43-s − 0.590·45-s + 0.845·47-s + 0.142·49-s − 0.553·51-s − 0.346·53-s − 0.492·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507240245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507240245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 - 0.870T + 19T^{2} \) |
| 23 | \( 1 - 0.402T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 - 0.727T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 - 0.237T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 6.34T + 97T^{2} \) |
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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
−9.716062957257656556130921449037, −8.804533265595839187605630242132, −7.978672568454589361369437251359, −6.74396373417567529861898778573, −6.15598876614160797555771619298, −5.50583879334216304352412444290, −4.79584939103530829218158918263, −3.29531561336803501942851340569, −2.34914795982490296689970687389, −0.929262005669110901447638003194,
0.929262005669110901447638003194, 2.34914795982490296689970687389, 3.29531561336803501942851340569, 4.79584939103530829218158918263, 5.50583879334216304352412444290, 6.15598876614160797555771619298, 6.74396373417567529861898778573, 7.978672568454589361369437251359, 8.804533265595839187605630242132, 9.716062957257656556130921449037
