L(s) = 1 | + 0.885·3-s − 0.282·5-s − 2.95·7-s − 2.21·9-s − 5.51·11-s + 6.01·13-s − 0.249·15-s − 3.87·17-s + 2.94·19-s − 2.61·21-s + 7.16·23-s − 4.92·25-s − 4.61·27-s + 8.17·29-s + 8.64·31-s − 4.88·33-s + 0.832·35-s − 7.02·37-s + 5.32·39-s − 4.63·41-s − 1.94·43-s + 0.624·45-s − 6.63·47-s + 1.70·49-s − 3.43·51-s − 5.95·53-s + 1.55·55-s + ⋯ |
L(s) = 1 | + 0.511·3-s − 0.126·5-s − 1.11·7-s − 0.738·9-s − 1.66·11-s + 1.66·13-s − 0.0645·15-s − 0.940·17-s + 0.675·19-s − 0.570·21-s + 1.49·23-s − 0.984·25-s − 0.889·27-s + 1.51·29-s + 1.55·31-s − 0.850·33-s + 0.140·35-s − 1.15·37-s + 0.853·39-s − 0.724·41-s − 0.296·43-s + 0.0931·45-s − 0.968·47-s + 0.243·49-s − 0.480·51-s − 0.817·53-s + 0.209·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.445117633\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445117633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 0.885T + 3T^{2} \) |
| 5 | \( 1 + 0.282T + 5T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 6.01T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 - 0.476T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−8.393278639340923998699271938428, −8.006748599977550639578609587166, −6.83756038864062116734422137373, −6.36428856627440116456030601526, −5.48012428476650680094190593034, −4.74023181612183537559109219489, −3.39329312006070214171753591318, −3.17081493831331946772935538168, −2.22950461839487177406252041717, −0.64151568683327839669083988418,
0.64151568683327839669083988418, 2.22950461839487177406252041717, 3.17081493831331946772935538168, 3.39329312006070214171753591318, 4.74023181612183537559109219489, 5.48012428476650680094190593034, 6.36428856627440116456030601526, 6.83756038864062116734422137373, 8.006748599977550639578609587166, 8.393278639340923998699271938428