| L(s) = 1 | − 0.612·3-s − 5-s − 7-s − 2.62·9-s + 5.91·11-s − 1.93·13-s + 0.612·15-s − 1.73·17-s + 4.81·19-s + 0.612·21-s + 2.86·23-s + 25-s + 3.44·27-s − 4.67·29-s + 3.67·31-s − 3.61·33-s + 35-s − 1.26·37-s + 1.18·39-s − 6.58·41-s − 43-s + 2.62·45-s − 4.00·47-s + 49-s + 1.06·51-s − 10.1·53-s − 5.91·55-s + ⋯ |
| L(s) = 1 | − 0.353·3-s − 0.447·5-s − 0.377·7-s − 0.875·9-s + 1.78·11-s − 0.536·13-s + 0.158·15-s − 0.421·17-s + 1.10·19-s + 0.133·21-s + 0.596·23-s + 0.200·25-s + 0.662·27-s − 0.868·29-s + 0.659·31-s − 0.630·33-s + 0.169·35-s − 0.208·37-s + 0.189·39-s − 1.02·41-s − 0.152·43-s + 0.391·45-s − 0.583·47-s + 0.142·49-s + 0.149·51-s − 1.40·53-s − 0.797·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277257482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277257482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.612T + 3T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 4.67T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 - 0.451T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.98T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 9.15T + 97T^{2} \) |
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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
−8.110898783860163734383652367646, −7.20339730335359544287888833510, −6.67949330196077778678416062977, −6.04003997923993378425005051349, −5.19500800499155197722184419424, −4.48538613310570680505253642154, −3.53649627805258855117512708720, −3.01989081277485596514036130974, −1.73029252020359985931412890542, −0.61013992884617315521054491242,
0.61013992884617315521054491242, 1.73029252020359985931412890542, 3.01989081277485596514036130974, 3.53649627805258855117512708720, 4.48538613310570680505253642154, 5.19500800499155197722184419424, 6.04003997923993378425005051349, 6.67949330196077778678416062977, 7.20339730335359544287888833510, 8.110898783860163734383652367646
