| L(s) = 1 | + 9·3-s + 31.9·5-s + 81·9-s + 260.·11-s − 769.·13-s + 287.·15-s + 1.55e3·17-s − 750.·19-s + 754.·23-s − 2.10e3·25-s + 729·27-s + 6.00e3·29-s + 6.42e3·31-s + 2.34e3·33-s − 4.77e3·37-s − 6.92e3·39-s + 5.42e3·41-s − 1.18e4·43-s + 2.58e3·45-s + 1.74e4·47-s + 1.39e4·51-s + 3.76e4·53-s + 8.33e3·55-s − 6.75e3·57-s − 2.20e4·59-s + 8.17e3·61-s − 2.46e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.571·5-s + 0.333·9-s + 0.650·11-s − 1.26·13-s + 0.330·15-s + 1.30·17-s − 0.476·19-s + 0.297·23-s − 0.673·25-s + 0.192·27-s + 1.32·29-s + 1.19·31-s + 0.375·33-s − 0.573·37-s − 0.729·39-s + 0.503·41-s − 0.981·43-s + 0.190·45-s + 1.15·47-s + 0.752·51-s + 1.84·53-s + 0.371·55-s − 0.275·57-s − 0.825·59-s + 0.281·61-s − 0.722·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.284377332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.284377332\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 31.9T + 3.12e3T^{2} \) |
| 11 | \( 1 - 260.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 769.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 750.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 754.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.20e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.17e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.30e4T + 8.58e9T^{2} \) |
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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
−9.934892363506763234647149182041, −9.113991791316503223734671704512, −8.188000831983854383773409715614, −7.29201958656919184321933107951, −6.36598688716242558748899118855, −5.28771184342674708876054141718, −4.25695826403004166418053211987, −3.04909450570473859424203583454, −2.09448299306356188416019700470, −0.879963474089920637508027276258,
0.879963474089920637508027276258, 2.09448299306356188416019700470, 3.04909450570473859424203583454, 4.25695826403004166418053211987, 5.28771184342674708876054141718, 6.36598688716242558748899118855, 7.29201958656919184321933107951, 8.188000831983854383773409715614, 9.113991791316503223734671704512, 9.934892363506763234647149182041
