| L(s) = 1 | + 0.221·3-s + 4.40·5-s + 5.16·7-s − 2.95·9-s + 3.46·11-s + 0.284·13-s + 0.975·15-s + 1.31·17-s − 7.10·19-s + 1.14·21-s − 4.02·23-s + 14.3·25-s − 1.31·27-s − 1.44·29-s + 4.42·31-s + 0.768·33-s + 22.7·35-s − 11.4·37-s + 0.0630·39-s + 5.15·41-s + 3.72·43-s − 12.9·45-s + 7.61·47-s + 19.7·49-s + 0.290·51-s + 1.44·53-s + 15.2·55-s + ⋯ |
| L(s) = 1 | + 0.127·3-s + 1.96·5-s + 1.95·7-s − 0.983·9-s + 1.04·11-s + 0.0789·13-s + 0.251·15-s + 0.317·17-s − 1.63·19-s + 0.249·21-s − 0.838·23-s + 2.87·25-s − 0.253·27-s − 0.268·29-s + 0.794·31-s + 0.133·33-s + 3.84·35-s − 1.88·37-s + 0.0100·39-s + 0.804·41-s + 0.568·43-s − 1.93·45-s + 1.11·47-s + 2.81·49-s + 0.0406·51-s + 0.198·53-s + 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.744462420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.744462420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.221T + 3T^{2} \) |
| 5 | \( 1 - 4.40T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.284T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 0.972T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 4.95T + 79T^{2} \) |
| 83 | \( 1 - 6.33T + 83T^{2} \) |
| 89 | \( 1 + 9.69T + 89T^{2} \) |
| 97 | \( 1 + 4.58T + 97T^{2} \) |
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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
−8.721900745616734477525824093262, −7.902058411195249523051892478574, −6.80847684119012680055516844745, −6.05703389163900363031030053468, −5.57444600486931132352185139558, −4.83746824359449219448103271764, −3.98938741303607841838281579491, −2.52742633062379321488518749961, −1.97020760108675343480343212747, −1.25409749383996920558997028694,
1.25409749383996920558997028694, 1.97020760108675343480343212747, 2.52742633062379321488518749961, 3.98938741303607841838281579491, 4.83746824359449219448103271764, 5.57444600486931132352185139558, 6.05703389163900363031030053468, 6.80847684119012680055516844745, 7.902058411195249523051892478574, 8.721900745616734477525824093262
