| L(s) = 1 | − 4.12·2-s + 9·4-s + 13.4·5-s + 31.4·7-s − 4.12·8-s − 55.4·10-s + 40.4·11-s − 129.·14-s − 55·16-s + 43.1·17-s + 26.9·19-s + 120.·20-s − 166.·22-s + 19.0·23-s + 55.6·25-s + 282.·28-s + 154.·29-s − 308.·31-s + 259.·32-s − 177.·34-s + 422.·35-s − 43.5·37-s − 111.·38-s − 55.4·40-s + 47.8·41-s + 342.·43-s + 364.·44-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.12·4-s + 1.20·5-s + 1.69·7-s − 0.182·8-s − 1.75·10-s + 1.11·11-s − 2.47·14-s − 0.859·16-s + 0.615·17-s + 0.325·19-s + 1.35·20-s − 1.61·22-s + 0.172·23-s + 0.445·25-s + 1.90·28-s + 0.986·29-s − 1.78·31-s + 1.43·32-s − 0.897·34-s + 2.03·35-s − 0.193·37-s − 0.474·38-s − 0.219·40-s + 0.182·41-s + 1.21·43-s + 1.24·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.015456328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015456328\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.12T + 8T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 - 31.4T + 343T^{2} \) |
| 11 | \( 1 - 40.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 43.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 43.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 590.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−9.012746474866907358567203225182, −8.579781852665699248962982321922, −7.63528215122639645499362559624, −7.03025156431203266532408067350, −5.91109963343907352625484927131, −5.13320023606361229951509780997, −4.06942430857133248660673100233, −2.35804987782895740355226355379, −1.56371757947904962003783036961, −0.991430142294612427054925713228,
0.991430142294612427054925713228, 1.56371757947904962003783036961, 2.35804987782895740355226355379, 4.06942430857133248660673100233, 5.13320023606361229951509780997, 5.91109963343907352625484927131, 7.03025156431203266532408067350, 7.63528215122639645499362559624, 8.579781852665699248962982321922, 9.012746474866907358567203225182
