| L(s) = 1 | − 3-s + 3.41·5-s + 9-s − 2·11-s + 2.58·13-s − 3.41·15-s + 2.24·17-s + 2.82·19-s + 7.65·23-s + 6.65·25-s − 27-s + 6.82·29-s − 1.17·31-s + 2·33-s + 4·37-s − 2.58·39-s − 6.24·41-s + 5.65·43-s + 3.41·45-s − 2.82·47-s − 2.24·51-s + 2·53-s − 6.82·55-s − 2.82·57-s + 1.17·59-s + 12.2·61-s + 8.82·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.52·5-s + 0.333·9-s − 0.603·11-s + 0.717·13-s − 0.881·15-s + 0.543·17-s + 0.648·19-s + 1.59·23-s + 1.33·25-s − 0.192·27-s + 1.26·29-s − 0.210·31-s + 0.348·33-s + 0.657·37-s − 0.414·39-s − 0.974·41-s + 0.862·43-s + 0.508·45-s − 0.412·47-s − 0.314·51-s + 0.274·53-s − 0.920·55-s − 0.374·57-s + 0.152·59-s + 1.56·61-s + 1.09·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.747597132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.747597132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−7.52210802750085118502778396137, −6.90888980573677251149018700657, −6.19115728853326387461163803000, −5.65425181736512599283611003488, −5.16796583272827472895508471260, −4.45244362565634792958744497950, −3.23002988698946517280984004325, −2.63327143219868689497359196856, −1.55846818512334558511461733326, −0.893221768593341318894201469004,
0.893221768593341318894201469004, 1.55846818512334558511461733326, 2.63327143219868689497359196856, 3.23002988698946517280984004325, 4.45244362565634792958744497950, 5.16796583272827472895508471260, 5.65425181736512599283611003488, 6.19115728853326387461163803000, 6.90888980573677251149018700657, 7.52210802750085118502778396137
