| L(s) = 1 | + 1.55·2-s − 5.59·4-s − 9.81·5-s − 21.0·8-s − 15.2·10-s + 70.6·11-s − 55.5·13-s + 12.0·16-s − 13.4·17-s − 91.3·19-s + 54.8·20-s + 109.·22-s − 113.·23-s − 28.7·25-s − 86.1·26-s − 12.4·29-s − 222.·31-s + 187.·32-s − 20.8·34-s + 257.·37-s − 141.·38-s + 206.·40-s − 286.·41-s + 4.81·43-s − 394.·44-s − 176.·46-s + 609.·47-s + ⋯ |
| L(s) = 1 | + 0.548·2-s − 0.699·4-s − 0.877·5-s − 0.931·8-s − 0.481·10-s + 1.93·11-s − 1.18·13-s + 0.188·16-s − 0.191·17-s − 1.10·19-s + 0.613·20-s + 1.06·22-s − 1.03·23-s − 0.229·25-s − 0.650·26-s − 0.0800·29-s − 1.29·31-s + 1.03·32-s − 0.104·34-s + 1.14·37-s − 0.605·38-s + 0.817·40-s − 1.09·41-s + 0.0170·43-s − 1.35·44-s − 0.565·46-s + 1.89·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.182476888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182476888\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 8T^{2} \) |
| 5 | \( 1 + 9.81T + 125T^{2} \) |
| 11 | \( 1 - 70.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 4.81T + 7.95e4T^{2} \) |
| 47 | \( 1 - 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 691.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 217.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 98.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 921.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 9.42T + 4.93e5T^{2} \) |
| 83 | \( 1 - 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 253.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 59.2T + 9.12e5T^{2} \) |
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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
−9.268491110835052559365642046344, −8.504644085997278823099769013714, −7.65539286313107260325773739977, −6.69081232173316504816994556883, −5.91351902338233832174905264054, −4.74994257986462456538914680011, −4.07000984457363537420500899047, −3.57523787172336859884453511744, −2.06755728409460973295178542563, −0.49306999778494320234740779790,
0.49306999778494320234740779790, 2.06755728409460973295178542563, 3.57523787172336859884453511744, 4.07000984457363537420500899047, 4.74994257986462456538914680011, 5.91351902338233832174905264054, 6.69081232173316504816994556883, 7.65539286313107260325773739977, 8.504644085997278823099769013714, 9.268491110835052559365642046344
