| L(s) = 1 | − 3-s + 1.51·5-s + 7-s + 9-s − 6.20·11-s + 13-s − 1.51·15-s + 4·17-s + 5.71·19-s − 21-s + 4.68·23-s − 2.71·25-s − 27-s − 3.71·29-s − 6.68·31-s + 6.20·33-s + 1.51·35-s + 5.02·37-s − 39-s + 7.22·41-s − 10.6·43-s + 1.51·45-s + 4.53·47-s + 49-s − 4·51-s + 0.688·53-s − 9.37·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.676·5-s + 0.377·7-s + 0.333·9-s − 1.86·11-s + 0.277·13-s − 0.390·15-s + 0.970·17-s + 1.31·19-s − 0.218·21-s + 0.977·23-s − 0.542·25-s − 0.192·27-s − 0.689·29-s − 1.20·31-s + 1.07·33-s + 0.255·35-s + 0.826·37-s − 0.160·39-s + 1.12·41-s − 1.62·43-s + 0.225·45-s + 0.661·47-s + 0.142·49-s − 0.560·51-s + 0.0945·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.721525652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.721525652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 - 0.688T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.02T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 5.66T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−8.108921916518374578837235574750, −7.69602719107383748666511536760, −6.98521649414171493670516596825, −5.89856802903626834042288315361, −5.31361194746357459388088041936, −5.10902111974452141061661994333, −3.76937172643275383597147251935, −2.84783008885730966503396804295, −1.90198622090479394331785473645, −0.76708973085246373284997917014,
0.76708973085246373284997917014, 1.90198622090479394331785473645, 2.84783008885730966503396804295, 3.76937172643275383597147251935, 5.10902111974452141061661994333, 5.31361194746357459388088041936, 5.89856802903626834042288315361, 6.98521649414171493670516596825, 7.69602719107383748666511536760, 8.108921916518374578837235574750
