| L(s) = 1 | + 1.41·3-s − 4.24·7-s − 0.999·9-s − 5.65·11-s − 2·13-s − 6·17-s + 2.82·19-s − 6·21-s + 7.07·23-s − 5.65·27-s + 4·29-s − 2.82·31-s − 8.00·33-s + 2·37-s − 2.82·39-s + 8·41-s − 1.41·43-s + 1.41·47-s + 10.9·49-s − 8.48·51-s − 2·53-s + 4.00·57-s − 2.82·59-s + 14·61-s + 4.24·63-s − 4.24·67-s + 10.0·69-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 1.60·7-s − 0.333·9-s − 1.70·11-s − 0.554·13-s − 1.45·17-s + 0.648·19-s − 1.30·21-s + 1.47·23-s − 1.08·27-s + 0.742·29-s − 0.508·31-s − 1.39·33-s + 0.328·37-s − 0.452·39-s + 1.24·41-s − 0.215·43-s + 0.206·47-s + 1.57·49-s − 1.18·51-s − 0.274·53-s + 0.529·57-s − 0.368·59-s + 1.79·61-s + 0.534·63-s − 0.518·67-s + 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061990463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061990463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.074086687464224393596966877686, −7.28396520912275412880506758914, −6.80834289245399036469638759896, −5.88096670538898794673786482450, −5.22417398028819246446618043501, −4.33700118840588582943993187673, −3.25570041921570547737112506869, −2.80725815646059871585639355420, −2.30402609223903457218658652078, −0.46997589520305112521014395683,
0.46997589520305112521014395683, 2.30402609223903457218658652078, 2.80725815646059871585639355420, 3.25570041921570547737112506869, 4.33700118840588582943993187673, 5.22417398028819246446618043501, 5.88096670538898794673786482450, 6.80834289245399036469638759896, 7.28396520912275412880506758914, 8.074086687464224393596966877686
