| L(s) = 1 | − 2.74·3-s + 2.14·5-s − 3.54·7-s + 4.53·9-s + 6.54·11-s + 5.92·13-s − 5.90·15-s + 3.02·17-s − 5.44·19-s + 9.74·21-s − 2.97·23-s − 0.379·25-s − 4.21·27-s − 7.08·29-s − 2.90·31-s − 17.9·33-s − 7.63·35-s + 5.80·37-s − 16.2·39-s − 8.06·41-s + 8.02·43-s + 9.74·45-s + 12.1·47-s + 5.60·49-s − 8.31·51-s − 4.41·53-s + 14.0·55-s + ⋯ |
| L(s) = 1 | − 1.58·3-s + 0.961·5-s − 1.34·7-s + 1.51·9-s + 1.97·11-s + 1.64·13-s − 1.52·15-s + 0.734·17-s − 1.24·19-s + 2.12·21-s − 0.620·23-s − 0.0758·25-s − 0.810·27-s − 1.31·29-s − 0.521·31-s − 3.12·33-s − 1.28·35-s + 0.954·37-s − 2.60·39-s − 1.25·41-s + 1.22·43-s + 1.45·45-s + 1.77·47-s + 0.800·49-s − 1.16·51-s − 0.605·53-s + 1.89·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172813772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172813772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 6.54T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.41T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.367742828829231342575841474943, −8.648413499035737963851364268925, −7.11533725101647248220663045477, −6.39875637329454409301493178247, −6.03081814230769498512376816072, −5.66354027884509242359397896544, −4.12665620990625741877177747089, −3.64742649786835247462259709571, −1.85397193687510446504741540830, −0.814853203175159875830874087074,
0.814853203175159875830874087074, 1.85397193687510446504741540830, 3.64742649786835247462259709571, 4.12665620990625741877177747089, 5.66354027884509242359397896544, 6.03081814230769498512376816072, 6.39875637329454409301493178247, 7.11533725101647248220663045477, 8.648413499035737963851364268925, 9.367742828829231342575841474943
