| L(s) = 1 | − 10.6·5-s + 6.65·11-s − 75.9·13-s − 104.·17-s − 85.4·19-s + 68.6·23-s − 11.4·25-s − 87.7·29-s − 62.7·31-s + 42.2·37-s + 313.·41-s + 306.·43-s + 215.·47-s − 525.·53-s − 70.8·55-s + 360.·59-s − 800.·61-s + 809.·65-s − 40.2·67-s + 298.·71-s − 517.·73-s − 1.22e3·79-s + 1.32e3·83-s + 1.11e3·85-s − 639.·89-s + 910.·95-s − 1.42e3·97-s + ⋯ |
| L(s) = 1 | − 0.953·5-s + 0.182·11-s − 1.62·13-s − 1.48·17-s − 1.03·19-s + 0.622·23-s − 0.0917·25-s − 0.562·29-s − 0.363·31-s + 0.187·37-s + 1.19·41-s + 1.08·43-s + 0.667·47-s − 1.36·53-s − 0.173·55-s + 0.795·59-s − 1.68·61-s + 1.54·65-s − 0.0733·67-s + 0.499·71-s − 0.829·73-s − 1.74·79-s + 1.75·83-s + 1.41·85-s − 0.762·89-s + 0.983·95-s − 1.49·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6676237293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6676237293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 10.6T + 125T^{2} \) |
| 11 | \( 1 - 6.65T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 42.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 800.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 40.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 298.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 517.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−9.014010758221168439885918234740, −8.044270543212768262366529282361, −7.37221018423663699644589320327, −6.74443901098487182520780942117, −5.69560608933825795733586324940, −4.51447467953315895157740228742, −4.21182511384713283381241760971, −2.89096471932008851650028836790, −2.00959072363156848984600150792, −0.36493675076773575246073066020,
0.36493675076773575246073066020, 2.00959072363156848984600150792, 2.89096471932008851650028836790, 4.21182511384713283381241760971, 4.51447467953315895157740228742, 5.69560608933825795733586324940, 6.74443901098487182520780942117, 7.37221018423663699644589320327, 8.044270543212768262366529282361, 9.014010758221168439885918234740
