| L(s) = 1 | + 5.21·2-s + 19.2·4-s − 5.83·5-s + 31.3·7-s + 58.6·8-s − 30.4·10-s − 16.2·11-s + 163.·14-s + 152.·16-s + 54·17-s − 66.3·19-s − 112.·20-s − 84.9·22-s + 182.·23-s − 90.9·25-s + 602.·28-s + 164.·29-s + 58.9·31-s + 325.·32-s + 281.·34-s − 182.·35-s + 110.·37-s − 346.·38-s − 342.·40-s + 55.0·41-s − 113.·43-s − 313.·44-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.40·4-s − 0.522·5-s + 1.69·7-s + 2.59·8-s − 0.963·10-s − 0.446·11-s + 3.11·14-s + 2.37·16-s + 0.770·17-s − 0.801·19-s − 1.25·20-s − 0.823·22-s + 1.65·23-s − 0.727·25-s + 4.06·28-s + 1.05·29-s + 0.341·31-s + 1.79·32-s + 1.42·34-s − 0.883·35-s + 0.490·37-s − 1.47·38-s − 1.35·40-s + 0.209·41-s − 0.401·43-s − 1.07·44-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
| \(L(2)\) |
\(\approx\) |
\(8.737597010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.737597010\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.21T + 8T^{2} \) |
| 5 | \( 1 + 5.83T + 125T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 + 16.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 55.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 514.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 265.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 468.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 852.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 574.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 66.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−8.874009475902276978685349659665, −7.84922908714090312543868154871, −7.49348972864370184736481706121, −6.39760601505430598532929757638, −5.50181541502453252223599188487, −4.76605052051313146434766120682, −4.33345728255767385996132094379, −3.23048150081354990052309841992, −2.30695114666928373366558088683, −1.20828584253536194609831584715,
1.20828584253536194609831584715, 2.30695114666928373366558088683, 3.23048150081354990052309841992, 4.33345728255767385996132094379, 4.76605052051313146434766120682, 5.50181541502453252223599188487, 6.39760601505430598532929757638, 7.49348972864370184736481706121, 7.84922908714090312543868154871, 8.874009475902276978685349659665
