| L(s) = 1 | − 1.95·3-s + 1.53·5-s − 0.941·7-s + 0.828·9-s + 2.33·11-s − 0.0783·13-s − 2.99·15-s − 0.988·17-s − 3.58·19-s + 1.84·21-s − 2.65·25-s + 4.24·27-s + 5.83·29-s − 5.35·31-s − 4.57·33-s − 1.44·35-s + 8.82·37-s + 0.153·39-s + 9.58·41-s − 6.50·43-s + 1.26·45-s − 1.33·47-s − 6.11·49-s + 1.93·51-s + 6.41·53-s + 3.57·55-s + 7.00·57-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 0.684·5-s − 0.355·7-s + 0.276·9-s + 0.704·11-s − 0.0217·13-s − 0.773·15-s − 0.239·17-s − 0.821·19-s + 0.401·21-s − 0.531·25-s + 0.817·27-s + 1.08·29-s − 0.961·31-s − 0.796·33-s − 0.243·35-s + 1.45·37-s + 0.0245·39-s + 1.49·41-s − 0.992·43-s + 0.189·45-s − 0.195·47-s − 0.873·49-s + 0.270·51-s + 0.881·53-s + 0.482·55-s + 0.928·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164180798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164180798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 0.941T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 + 0.0783T + 13T^{2} \) |
| 17 | \( 1 + 0.988T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 6.86T + 73T^{2} \) |
| 79 | \( 1 - 0.850T + 79T^{2} \) |
| 83 | \( 1 - 5.27T + 83T^{2} \) |
| 89 | \( 1 - 7.14T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−8.499816388812341840946471718821, −7.52190030942951340602073363207, −6.55671110178136825449559386810, −6.23792181717159986224754072226, −5.62608355525197253010627246787, −4.75022737429366148817077525876, −4.01936487519004190237669828317, −2.85270049170639537337429286093, −1.82936560700028761835436614657, −0.64575997440847504015086996864,
0.64575997440847504015086996864, 1.82936560700028761835436614657, 2.85270049170639537337429286093, 4.01936487519004190237669828317, 4.75022737429366148817077525876, 5.62608355525197253010627246787, 6.23792181717159986224754072226, 6.55671110178136825449559386810, 7.52190030942951340602073363207, 8.499816388812341840946471718821
