| L(s) = 1 | − 4.91·2-s + 16.1·4-s + 5·5-s − 39.8·8-s − 24.5·10-s − 12.0·11-s − 46.9·13-s + 66.8·16-s + 44.7·17-s + 31.7·19-s + 80.5·20-s + 58.9·22-s + 39.4·23-s + 25·25-s + 230.·26-s − 87.2·29-s − 304.·31-s − 9.27·32-s − 219.·34-s − 151.·37-s − 155.·38-s − 199.·40-s + 282.·41-s − 143.·43-s − 193.·44-s − 193.·46-s + 88.8·47-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s + 0.447·5-s − 1.76·8-s − 0.776·10-s − 0.328·11-s − 1.00·13-s + 1.04·16-s + 0.637·17-s + 0.383·19-s + 0.900·20-s + 0.571·22-s + 0.357·23-s + 0.200·25-s + 1.73·26-s − 0.558·29-s − 1.76·31-s − 0.0512·32-s − 1.10·34-s − 0.671·37-s − 0.665·38-s − 0.787·40-s + 1.07·41-s − 0.507·43-s − 0.662·44-s − 0.620·46-s + 0.275·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6972964547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6972964547\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.91T + 8T^{2} \) |
| 11 | \( 1 + 12.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 88.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 712.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 65.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 698.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 476.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 256.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.14e3T + 9.12e5T^{2} \) |
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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
−8.960355759356425471634744498266, −7.84350047198862292184562425952, −7.50373830095534241989549281522, −6.72238697308052325428664707840, −5.74250800780017887260632164231, −4.92793998481847938516708962847, −3.41136702956808148589381989914, −2.38711252473695870567400913308, −1.61049076730129647178726433636, −0.49161124504479331312284023424,
0.49161124504479331312284023424, 1.61049076730129647178726433636, 2.38711252473695870567400913308, 3.41136702956808148589381989914, 4.92793998481847938516708962847, 5.74250800780017887260632164231, 6.72238697308052325428664707840, 7.50373830095534241989549281522, 7.84350047198862292184562425952, 8.960355759356425471634744498266
