| L(s) = 1 | + 3-s + 3.41·5-s + 0.585·7-s + 9-s + 2·11-s − 2.82·13-s + 3.41·15-s − 7.65·17-s + 5.65·19-s + 0.585·21-s + 6.82·23-s + 6.65·25-s + 27-s + 3.41·29-s + 7.41·31-s + 2·33-s + 2·35-s + 1.65·37-s − 2.82·39-s − 0.343·41-s − 9.65·43-s + 3.41·45-s − 4.48·47-s − 6.65·49-s − 7.65·51-s − 7.89·53-s + 6.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.52·5-s + 0.221·7-s + 0.333·9-s + 0.603·11-s − 0.784·13-s + 0.881·15-s − 1.85·17-s + 1.29·19-s + 0.127·21-s + 1.42·23-s + 1.33·25-s + 0.192·27-s + 0.634·29-s + 1.33·31-s + 0.348·33-s + 0.338·35-s + 0.272·37-s − 0.452·39-s − 0.0535·41-s − 1.47·43-s + 0.508·45-s − 0.654·47-s − 0.950·49-s − 1.07·51-s − 1.08·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.887457950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.887457950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−9.488508849873647533666593744900, −8.873292555372058290581531830652, −7.973045547927781813057653953377, −6.74408826270721560195652484993, −6.50388636809120226078842110448, −5.14104286937328051225299031907, −4.65874669023762701943332639827, −3.14235478433949288504287554723, −2.32747443667696939674509316947, −1.34299784903716846226126702258,
1.34299784903716846226126702258, 2.32747443667696939674509316947, 3.14235478433949288504287554723, 4.65874669023762701943332639827, 5.14104286937328051225299031907, 6.50388636809120226078842110448, 6.74408826270721560195652484993, 7.973045547927781813057653953377, 8.873292555372058290581531830652, 9.488508849873647533666593744900
