L(s) = 1 | + 5-s + 4.24·11-s + 3.24·13-s − 4.24·17-s − 7·19-s + 4.24·23-s + 25-s + 1.75·29-s − 9.48·31-s + 3.24·37-s + 4.24·41-s + 3.24·43-s + 6·47-s − 8.48·53-s + 4.24·55-s + 10.2·59-s + 4.48·61-s + 3.24·65-s − 5.24·67-s + 12.7·71-s + 9.24·73-s + 11·79-s + 10.2·83-s − 4.24·85-s + 10.2·89-s − 7·95-s − 0.485·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.27·11-s + 0.899·13-s − 1.02·17-s − 1.60·19-s + 0.884·23-s + 0.200·25-s + 0.326·29-s − 1.70·31-s + 0.533·37-s + 0.662·41-s + 0.494·43-s + 0.875·47-s − 1.16·53-s + 0.572·55-s + 1.33·59-s + 0.574·61-s + 0.402·65-s − 0.640·67-s + 1.51·71-s + 1.08·73-s + 1.23·79-s + 1.12·83-s − 0.460·85-s + 1.08·89-s − 0.718·95-s − 0.0492·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.461439560\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.461439560\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 0.485T + 97T^{2} \) |
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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
−7.79076275214795887333765216091, −6.69934425875923768946988774096, −6.58880000276517616250572309187, −5.82611006553392563817109859347, −4.96171545632581238238031580895, −4.09161669522373807888201651537, −3.69011076642521755905085747470, −2.47455387140088110597940996929, −1.79209268752176196911526702585, −0.77929302375563234563813573220,
0.77929302375563234563813573220, 1.79209268752176196911526702585, 2.47455387140088110597940996929, 3.69011076642521755905085747470, 4.09161669522373807888201651537, 4.96171545632581238238031580895, 5.82611006553392563817109859347, 6.58880000276517616250572309187, 6.69934425875923768946988774096, 7.79076275214795887333765216091