| L(s) = 1 | − 1.83·2-s + 3-s + 1.35·4-s + 1.36·5-s − 1.83·6-s + 7-s + 1.17·8-s + 9-s − 2.50·10-s − 5.37·11-s + 1.35·12-s − 1.02·13-s − 1.83·14-s + 1.36·15-s − 4.87·16-s + 5.32·17-s − 1.83·18-s − 1.77·19-s + 1.85·20-s + 21-s + 9.84·22-s − 1.40·23-s + 1.17·24-s − 3.12·25-s + 1.87·26-s + 27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.678·4-s + 0.612·5-s − 0.747·6-s + 0.377·7-s + 0.416·8-s + 0.333·9-s − 0.793·10-s − 1.61·11-s + 0.391·12-s − 0.283·13-s − 0.489·14-s + 0.353·15-s − 1.21·16-s + 1.29·17-s − 0.431·18-s − 0.406·19-s + 0.415·20-s + 0.218·21-s + 2.09·22-s − 0.292·23-s + 0.240·24-s − 0.624·25-s + 0.367·26-s + 0.192·27-s + 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 3.44T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 9.83T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 2.35T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.73926089862318615884691971481, −7.18914627865928027185114981859, −6.28947551430085501721863239407, −5.28555762751202706242202175495, −4.88555906243099474246776285979, −3.73952547800430340621211603174, −2.71137495275103288621482656689, −2.08585752516804177427438833109, −1.24399653651461644689975653248, 0,
1.24399653651461644689975653248, 2.08585752516804177427438833109, 2.71137495275103288621482656689, 3.73952547800430340621211603174, 4.88555906243099474246776285979, 5.28555762751202706242202175495, 6.28947551430085501721863239407, 7.18914627865928027185114981859, 7.73926089862318615884691971481
