L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 2i·7-s − i·8-s − 9-s + 4·11-s + i·12-s − i·13-s − 2·14-s + 16-s − 4i·17-s − i·18-s + 2·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.277i·13-s − 0.534·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s + 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.730313377\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.730313377\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.333651699048819729581389333850, −8.396021183708912475172699077001, −7.49789540613956136445514367164, −7.01694950222565986529522763079, −5.98289809972298101134254239087, −5.58414138229537534019858177694, −4.46646521756106142852508223789, −3.41617392052549645280366711464, −2.26189837642423193082521316818, −0.899642979826364470409331637264,
0.932189896907463189511738829103, 2.11965407808596020444206751297, 3.52139867565181472307634028879, 3.96300462650630747637060208031, 4.77372062108164667041017629672, 5.87093037457966426350910039323, 6.70847123789544165730707120534, 7.69052931837315698966815837002, 8.603681277731557582636437643762, 9.293934863305609005390907939803