| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 88.0·5-s − 36·6-s + 63.3·7-s + 64·8-s + 81·9-s + 352.·10-s − 101.·11-s − 144·12-s − 228.·13-s + 253.·14-s − 792.·15-s + 256·16-s + 539.·17-s + 324·18-s + 985.·19-s + 1.40e3·20-s − 570.·21-s − 405.·22-s − 529·23-s − 576·24-s + 4.62e3·25-s − 912.·26-s − 729·27-s + 1.01e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.57·5-s − 0.408·6-s + 0.488·7-s + 0.353·8-s + 0.333·9-s + 1.11·10-s − 0.252·11-s − 0.288·12-s − 0.374·13-s + 0.345·14-s − 0.909·15-s + 0.250·16-s + 0.453·17-s + 0.235·18-s + 0.626·19-s + 0.787·20-s − 0.282·21-s − 0.178·22-s − 0.208·23-s − 0.204·24-s + 1.47·25-s − 0.264·26-s − 0.192·27-s + 0.244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.478520379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.478520379\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 88.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 63.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 101.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 228.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 539.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 985.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 337.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.48e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 790.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.90e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.68e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.69e4T + 8.58e9T^{2} \) |
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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
−12.39486522838029672630976800935, −11.33806278529903718963931983404, −10.27698754823944435298759204015, −9.483579717591860760579550174225, −7.77135368907234504094586297036, −6.39169041306561758706378699428, −5.59360062744815808607324870745, −4.65650328682685084618463800026, −2.67156239093926511181327621521, −1.33803217440459982490328474974,
1.33803217440459982490328474974, 2.67156239093926511181327621521, 4.65650328682685084618463800026, 5.59360062744815808607324870745, 6.39169041306561758706378699428, 7.77135368907234504094586297036, 9.483579717591860760579550174225, 10.27698754823944435298759204015, 11.33806278529903718963931983404, 12.39486522838029672630976800935
