| L(s) = 1 | + i·2-s − 4-s + (2.56 + 0.648i)7-s − i·8-s − 1.29i·11-s + 3.13i·13-s + (−0.648 + 2.56i)14-s + 16-s + 5.53·17-s − 7.37i·19-s + 1.29·22-s − 1.83i·23-s − 3.13·26-s + (−2.56 − 0.648i)28-s + 1.83i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.969 + 0.245i)7-s − 0.353i·8-s − 0.390i·11-s + 0.868i·13-s + (−0.173 + 0.685i)14-s + 0.250·16-s + 1.34·17-s − 1.69i·19-s + 0.276·22-s − 0.382i·23-s − 0.613·26-s + (−0.484 − 0.122i)28-s + 0.340i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014125658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014125658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.56 - 0.648i)T \) |
| good | 11 | \( 1 + 1.29iT - 11T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 7.37iT - 19T^{2} \) |
| 23 | \( 1 + 1.83iT - 23T^{2} \) |
| 29 | \( 1 - 1.83iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 4.42iT - 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 4.88iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 7.37iT - 71T^{2} \) |
| 73 | \( 1 - 3.40iT - 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
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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
−8.537310900084858749769378396334, −7.982795710818231639725453327525, −7.29664228162517516694778965913, −6.46459527812425708798317724480, −5.72968533825558750962528221471, −4.88698347486874672314477674820, −4.35287134287515595053302353765, −3.19193278162007376066262115585, −2.06238771837244040127523595561, −0.76447652619687584182515653060,
1.08898612107020500047483447233, 1.80928622493081156714518889660, 3.09557148168930825679279819238, 3.74663061951671635666622452976, 4.79592734900566179206976965521, 5.36681924551424969954378347512, 6.22018495807297174051101997979, 7.47386370069462294678182582764, 7.989821889050953579009372320702, 8.479081630445255267292416585877
