| L(s) = 1 | + 1.34·3-s + 5·5-s − 32.8·7-s − 25.1·9-s − 58.9·11-s − 60.5·13-s + 6.73·15-s + 31.1·17-s + 43.2·19-s − 44.2·21-s − 23·23-s + 25·25-s − 70.3·27-s − 13.6·29-s − 287.·31-s − 79.4·33-s − 164.·35-s + 279.·37-s − 81.5·39-s − 419.·41-s − 421.·43-s − 125.·45-s − 117.·47-s + 738.·49-s + 42.0·51-s + 439.·53-s − 294.·55-s + ⋯ |
| L(s) = 1 | + 0.259·3-s + 0.447·5-s − 1.77·7-s − 0.932·9-s − 1.61·11-s − 1.29·13-s + 0.115·15-s + 0.444·17-s + 0.522·19-s − 0.460·21-s − 0.208·23-s + 0.200·25-s − 0.501·27-s − 0.0870·29-s − 1.66·31-s − 0.418·33-s − 0.793·35-s + 1.24·37-s − 0.334·39-s − 1.59·41-s − 1.49·43-s − 0.417·45-s − 0.365·47-s + 2.15·49-s + 0.115·51-s + 1.13·53-s − 0.722·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4363987840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4363987840\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 1.34T + 27T^{2} \) |
| 7 | \( 1 + 32.8T + 343T^{2} \) |
| 11 | \( 1 + 58.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 13.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 421.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 439.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 248.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 888.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 755.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 648.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 542.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 685.T + 9.12e5T^{2} \) |
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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
−9.050378518248265790336173251752, −8.061122989213122962783575057427, −7.33443941042687868323054546131, −6.50624219920037333731454568552, −5.55270623413318353963805272217, −5.14530818275696867217114074330, −3.54137272880196648263579290232, −2.90504271293305082414956330423, −2.22724034943482714338734816417, −0.27709169418141042888344142564,
0.27709169418141042888344142564, 2.22724034943482714338734816417, 2.90504271293305082414956330423, 3.54137272880196648263579290232, 5.14530818275696867217114074330, 5.55270623413318353963805272217, 6.50624219920037333731454568552, 7.33443941042687868323054546131, 8.061122989213122962783575057427, 9.050378518248265790336173251752
