| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 4.73·7-s − 1.73·8-s + 1.46·11-s + 1.26·13-s + 8.19·14-s − 5·16-s − 1.46·17-s + 5.46·19-s + 2.53·22-s − 0.535·23-s + 2.19·26-s + 4.73·28-s − 0.732·29-s − 31-s − 5.19·32-s − 2.53·34-s + 6.73·37-s + 9.46·38-s − 3.46·41-s + 4·43-s + 1.46·44-s − 0.928·46-s − 6·47-s + 15.3·49-s + 1.26·52-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 1.78·7-s − 0.612·8-s + 0.441·11-s + 0.351·13-s + 2.19·14-s − 1.25·16-s − 0.355·17-s + 1.25·19-s + 0.540·22-s − 0.111·23-s + 0.430·26-s + 0.894·28-s − 0.135·29-s − 0.179·31-s − 0.918·32-s − 0.434·34-s + 1.10·37-s + 1.53·38-s − 0.541·41-s + 0.609·43-s + 0.220·44-s − 0.136·46-s − 0.875·47-s + 2.19·49-s + 0.175·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.894639987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.894639987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.893053687900416933476265502331, −7.16364360063009279840755122965, −6.36701859234604415630090117041, −5.44810167435516146425176805440, −5.20515842560853814391480133714, −4.31415015187014508343623516518, −3.90664209868322642203887338822, −2.88018990003401219217629670617, −1.97868750679910080220297006766, −0.998118981999872480378853620953,
0.998118981999872480378853620953, 1.97868750679910080220297006766, 2.88018990003401219217629670617, 3.90664209868322642203887338822, 4.31415015187014508343623516518, 5.20515842560853814391480133714, 5.44810167435516146425176805440, 6.36701859234604415630090117041, 7.16364360063009279840755122965, 7.893053687900416933476265502331
