L(s) = 1 | − 2-s + 0.259·3-s + 4-s + 5-s − 0.259·6-s − 4.66·7-s − 8-s − 2.93·9-s − 10-s − 3.14·11-s + 0.259·12-s − 5.01·13-s + 4.66·14-s + 0.259·15-s + 16-s − 6.42·17-s + 2.93·18-s − 5.69·19-s + 20-s − 1.20·21-s + 3.14·22-s − 0.259·24-s + 25-s + 5.01·26-s − 1.53·27-s − 4.66·28-s + 4.01·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.149·3-s + 0.5·4-s + 0.447·5-s − 0.105·6-s − 1.76·7-s − 0.353·8-s − 0.977·9-s − 0.316·10-s − 0.947·11-s + 0.0747·12-s − 1.38·13-s + 1.24·14-s + 0.0668·15-s + 0.250·16-s − 1.55·17-s + 0.691·18-s − 1.30·19-s + 0.223·20-s − 0.263·21-s + 0.670·22-s − 0.0528·24-s + 0.200·25-s + 0.982·26-s − 0.295·27-s − 0.881·28-s + 0.744·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06750400839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06750400839\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 0.259T + 3T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 - 0.125T + 31T^{2} \) |
| 37 | \( 1 - 0.835T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 4.91T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 8.75T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 5.16T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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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
−8.557035985359708747247635589831, −7.43192979557721503066827121452, −6.67036682901929036801547430838, −6.35190145602636264494736026354, −5.44888317796631195924778850332, −4.58762890087682913011508250668, −3.34723060766532297184981916945, −2.59341930704567337320829616033, −2.20208301209994307012826267470, −0.13794169379743533577565022554,
0.13794169379743533577565022554, 2.20208301209994307012826267470, 2.59341930704567337320829616033, 3.34723060766532297184981916945, 4.58762890087682913011508250668, 5.44888317796631195924778850332, 6.35190145602636264494736026354, 6.67036682901929036801547430838, 7.43192979557721503066827121452, 8.557035985359708747247635589831