L(s) = 1 | − 2.25·2-s − 3-s + 3.10·4-s + 2.25·6-s + 3.65·7-s − 2.48·8-s + 9-s − 5.39·11-s − 3.10·12-s + 3.76·13-s − 8.24·14-s − 0.583·16-s − 6.40·17-s − 2.25·18-s − 8.07·19-s − 3.65·21-s + 12.1·22-s − 4.76·23-s + 2.48·24-s − 8.50·26-s − 27-s + 11.3·28-s − 3.78·29-s − 5.26·31-s + 6.29·32-s + 5.39·33-s + 14.4·34-s + ⋯ |
L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.922·6-s + 1.38·7-s − 0.879·8-s + 0.333·9-s − 1.62·11-s − 0.895·12-s + 1.04·13-s − 2.20·14-s − 0.145·16-s − 1.55·17-s − 0.532·18-s − 1.85·19-s − 0.797·21-s + 2.59·22-s − 0.993·23-s + 0.507·24-s − 1.66·26-s − 0.192·27-s + 2.14·28-s − 0.702·29-s − 0.944·31-s + 1.11·32-s + 0.938·33-s + 2.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4565448840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4565448840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 53 | \( 1 + 9.62T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.493775410446760632947405609122, −8.011119521715102470383174390148, −7.44876091578675731734212005981, −6.48511976285172257360578883769, −5.79641226223031473750999038170, −4.77430376312352453949168613063, −4.10559786328867350870880364211, −2.23380511862968225243704453166, −1.93091815465900330252277677117, −0.50549684789553836895072413164,
0.50549684789553836895072413164, 1.93091815465900330252277677117, 2.23380511862968225243704453166, 4.10559786328867350870880364211, 4.77430376312352453949168613063, 5.79641226223031473750999038170, 6.48511976285172257360578883769, 7.44876091578675731734212005981, 8.011119521715102470383174390148, 8.493775410446760632947405609122