| L(s) = 1 | + 7.62·3-s + 11.9·5-s + 26.1·7-s + 31.2·9-s + 3.24·11-s + 20.0·13-s + 90.9·15-s − 17·17-s − 57.3·19-s + 199.·21-s + 77.0·23-s + 17.0·25-s + 32.1·27-s + 286.·29-s − 8.54·31-s + 24.7·33-s + 311.·35-s − 357.·37-s + 152.·39-s + 194.·41-s + 74.2·43-s + 372.·45-s + 23.6·47-s + 339.·49-s − 129.·51-s − 104.·53-s + 38.6·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.06·5-s + 1.41·7-s + 1.15·9-s + 0.0889·11-s + 0.427·13-s + 1.56·15-s − 0.242·17-s − 0.692·19-s + 2.07·21-s + 0.698·23-s + 0.136·25-s + 0.229·27-s + 1.83·29-s − 0.0495·31-s + 0.130·33-s + 1.50·35-s − 1.59·37-s + 0.628·39-s + 0.740·41-s + 0.263·43-s + 1.23·45-s + 0.0732·47-s + 0.989·49-s − 0.356·51-s − 0.270·53-s + 0.0947·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.563567352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.563567352\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 7.62T + 27T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 11 | \( 1 - 3.24T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.54T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 23.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 939.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 520.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 953.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 486.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 685.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.217622548239410746129922983400, −8.667975749755592732467941930769, −8.122567726213881924064891639277, −7.17720604109603494090984520249, −6.13792496312166487579460444974, −5.04708568815336146461729240936, −4.18911549362245743790378678935, −2.94877686275764844642189462970, −2.07348301063855394971599999847, −1.34867793004962149981114150939,
1.34867793004962149981114150939, 2.07348301063855394971599999847, 2.94877686275764844642189462970, 4.18911549362245743790378678935, 5.04708568815336146461729240936, 6.13792496312166487579460444974, 7.17720604109603494090984520249, 8.122567726213881924064891639277, 8.667975749755592732467941930769, 9.217622548239410746129922983400
