L(s) = 1 | − 3-s + 5-s − 3.20·7-s + 9-s − 4.50·11-s + 4.52·13-s − 15-s − 5.03·17-s − 0.977·19-s + 3.20·21-s + 0.658·23-s + 25-s − 27-s − 8.92·29-s + 3.82·31-s + 4.50·33-s − 3.20·35-s + 1.34·37-s − 4.52·39-s + 8.69·41-s + 6.73·43-s + 45-s − 7.39·47-s + 3.27·49-s + 5.03·51-s + 7.43·53-s − 4.50·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.21·7-s + 0.333·9-s − 1.35·11-s + 1.25·13-s − 0.258·15-s − 1.22·17-s − 0.224·19-s + 0.699·21-s + 0.137·23-s + 0.200·25-s − 0.192·27-s − 1.65·29-s + 0.687·31-s + 0.784·33-s − 0.541·35-s + 0.220·37-s − 0.724·39-s + 1.35·41-s + 1.02·43-s + 0.149·45-s − 1.07·47-s + 0.468·49-s + 0.704·51-s + 1.02·53-s − 0.607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9904725126\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9904725126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 + 0.977T + 19T^{2} \) |
| 23 | \( 1 - 0.658T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 1.34T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 61 | \( 1 - 2.97T + 61T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 + 6.32T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.539498512238151907868810868013, −7.59893822097831011670327176438, −6.84473267873689079182414950299, −6.05243414312225377067748381689, −5.77728124583622732515945099346, −4.73494147699848649331895649752, −3.86038025176631282431977655562, −2.91890324625364819660217129144, −2.02821137427013659434976783432, −0.56243946230516188437827841109,
0.56243946230516188437827841109, 2.02821137427013659434976783432, 2.91890324625364819660217129144, 3.86038025176631282431977655562, 4.73494147699848649331895649752, 5.77728124583622732515945099346, 6.05243414312225377067748381689, 6.84473267873689079182414950299, 7.59893822097831011670327176438, 8.539498512238151907868810868013