| L(s) = 1 | + 0.147·3-s − 3.84·5-s − 0.153·7-s − 2.97·9-s − 3.15·11-s + 4.79·13-s − 0.568·15-s + 17-s − 6.26·19-s − 0.0227·21-s − 9.39·23-s + 9.76·25-s − 0.884·27-s − 1.54·29-s + 2.57·31-s − 0.466·33-s + 0.591·35-s − 3.41·37-s + 0.709·39-s + 8.01·41-s − 9.29·43-s + 11.4·45-s − 8.10·47-s − 6.97·49-s + 0.147·51-s − 6.65·53-s + 12.1·55-s + ⋯ |
| L(s) = 1 | + 0.0853·3-s − 1.71·5-s − 0.0581·7-s − 0.992·9-s − 0.950·11-s + 1.33·13-s − 0.146·15-s + 0.242·17-s − 1.43·19-s − 0.00496·21-s − 1.95·23-s + 1.95·25-s − 0.170·27-s − 0.286·29-s + 0.461·31-s − 0.0811·33-s + 0.0999·35-s − 0.562·37-s + 0.113·39-s + 1.25·41-s − 1.41·43-s + 1.70·45-s − 1.18·47-s − 0.996·49-s + 0.0207·51-s − 0.914·53-s + 1.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3001918273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3001918273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.147T + 3T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 + 0.153T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 23 | \( 1 + 9.39T + 23T^{2} \) |
| 29 | \( 1 + 1.54T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 61 | \( 1 + 0.546T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 + 6.95T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−8.055237266988840316317877117873, −7.35776073762151732624358216966, −6.31532500894471402514070098178, −5.94343415460665275230004071770, −4.86270745362965945173969726529, −4.18769912828659012407326075231, −3.51747943509675009854137836050, −2.92915472451788557017842586809, −1.78840944319893863689304067997, −0.25765974541292961397410254925,
0.25765974541292961397410254925, 1.78840944319893863689304067997, 2.92915472451788557017842586809, 3.51747943509675009854137836050, 4.18769912828659012407326075231, 4.86270745362965945173969726529, 5.94343415460665275230004071770, 6.31532500894471402514070098178, 7.35776073762151732624358216966, 8.055237266988840316317877117873
