| L(s) = 1 | + 1.27·5-s − 1.11·7-s + 5.36·11-s + 0.160·13-s + 17-s + 7.36·19-s − 3.92·23-s − 3.38·25-s + 5.06·29-s − 3.06·31-s − 1.41·35-s + 5.06·37-s + 8.01·41-s + 1.18·43-s + 2.54·47-s − 5.76·49-s − 10.1·53-s + 6.81·55-s + 6.17·59-s − 10.7·61-s + 0.204·65-s + 15.8·67-s + 0.845·71-s − 3.95·73-s − 5.95·77-s + 2.56·79-s + 0.222·83-s + ⋯ |
| L(s) = 1 | + 0.568·5-s − 0.419·7-s + 1.61·11-s + 0.0445·13-s + 0.242·17-s + 1.68·19-s − 0.819·23-s − 0.676·25-s + 0.940·29-s − 0.550·31-s − 0.238·35-s + 0.833·37-s + 1.25·41-s + 0.180·43-s + 0.370·47-s − 0.823·49-s − 1.39·53-s + 0.919·55-s + 0.804·59-s − 1.37·61-s + 0.0253·65-s + 1.93·67-s + 0.100·71-s − 0.463·73-s − 0.678·77-s + 0.288·79-s + 0.0243·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.477280149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.477280149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 - 0.160T + 13T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 - 5.06T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 6.17T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 0.845T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 - 0.222T + 83T^{2} \) |
| 89 | \( 1 + 8.72T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−8.224498308591763932454799355142, −7.52853103985785809397882221018, −6.70494270583237950823017541355, −6.10386138821250588935781384994, −5.53189169470535493815092562702, −4.47487361253582653359916507610, −3.72204022773422395990315327807, −2.93875327930849934216406510843, −1.80123153949319324885959967794, −0.918510847933424602927354754692,
0.918510847933424602927354754692, 1.80123153949319324885959967794, 2.93875327930849934216406510843, 3.72204022773422395990315327807, 4.47487361253582653359916507610, 5.53189169470535493815092562702, 6.10386138821250588935781384994, 6.70494270583237950823017541355, 7.52853103985785809397882221018, 8.224498308591763932454799355142
