| L(s) = 1 | + 10.6·3-s + 25·5-s − 176.·7-s − 129.·9-s + 400.·11-s + 784.·13-s + 266.·15-s + 740.·17-s + 2.77e3·19-s − 1.88e3·21-s + 176.·23-s + 625·25-s − 3.97e3·27-s + 5.54e3·29-s − 5.73e3·31-s + 4.27e3·33-s − 4.42e3·35-s + 6.79e3·37-s + 8.36e3·39-s + 1.84e4·41-s + 1.40e4·43-s − 3.23e3·45-s − 2.33e4·47-s + 1.45e4·49-s + 7.90e3·51-s − 279.·53-s + 1.00e4·55-s + ⋯ |
| L(s) = 1 | + 0.684·3-s + 0.447·5-s − 1.36·7-s − 0.531·9-s + 0.999·11-s + 1.28·13-s + 0.305·15-s + 0.621·17-s + 1.76·19-s − 0.933·21-s + 0.0697·23-s + 0.200·25-s − 1.04·27-s + 1.22·29-s − 1.07·31-s + 0.683·33-s − 0.610·35-s + 0.816·37-s + 0.881·39-s + 1.71·41-s + 1.16·43-s − 0.237·45-s − 1.54·47-s + 0.863·49-s + 0.425·51-s − 0.0136·53-s + 0.446·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.528991044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528991044\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 10.6T + 243T^{2} \) |
| 7 | \( 1 + 176.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 400.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 784.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 740.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 176.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 279.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.64e4T + 8.58e9T^{2} \) |
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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
−12.07975360464770812151126621751, −10.95220219134249528393651436645, −9.489773337676562380502306665961, −9.264060079113955795403561552532, −7.913303193950932212528755545455, −6.53593447542888657198130587298, −5.71323593477514402601164881831, −3.70068791203062227374229359900, −2.91263334607638103075147555966, −1.06293573692992576253978492751,
1.06293573692992576253978492751, 2.91263334607638103075147555966, 3.70068791203062227374229359900, 5.71323593477514402601164881831, 6.53593447542888657198130587298, 7.913303193950932212528755545455, 9.264060079113955795403561552532, 9.489773337676562380502306665961, 10.95220219134249528393651436645, 12.07975360464770812151126621751
