| L(s) = 1 | − 2.59·2-s + 3-s + 4.75·4-s + 1.02·5-s − 2.59·6-s − 4.07·7-s − 7.17·8-s + 9-s − 2.66·10-s − 2.37·11-s + 4.75·12-s + 4.83·13-s + 10.5·14-s + 1.02·15-s + 9.12·16-s + 1.25·17-s − 2.59·18-s + 6.30·19-s + 4.88·20-s − 4.07·21-s + 6.17·22-s − 2.91·23-s − 7.17·24-s − 3.94·25-s − 12.5·26-s + 27-s − 19.3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.37·4-s + 0.459·5-s − 1.06·6-s − 1.53·7-s − 2.53·8-s + 0.333·9-s − 0.843·10-s − 0.715·11-s + 1.37·12-s + 1.34·13-s + 2.82·14-s + 0.265·15-s + 2.28·16-s + 0.303·17-s − 0.612·18-s + 1.44·19-s + 1.09·20-s − 0.888·21-s + 1.31·22-s − 0.607·23-s − 1.46·24-s − 0.789·25-s − 2.46·26-s + 0.192·27-s − 3.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8005410212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8005410212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 + 1.97T + 31T^{2} \) |
| 37 | \( 1 + 0.814T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 43 | \( 1 + 3.25T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 + 0.0575T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 - 0.895T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.31T + 83T^{2} \) |
| 89 | \( 1 + 7.16T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.943443406101927926315376475546, −7.35791282457735574780731841299, −6.77271899998939241109764826140, −5.94710943515493831138362791762, −5.56580684140531960510465015212, −3.74024245750083130437548735093, −3.27736596992171792328631803333, −2.41304583829667452077859214100, −1.61200776604278682476427613446, −0.56718439895401522781267904635,
0.56718439895401522781267904635, 1.61200776604278682476427613446, 2.41304583829667452077859214100, 3.27736596992171792328631803333, 3.74024245750083130437548735093, 5.56580684140531960510465015212, 5.94710943515493831138362791762, 6.77271899998939241109764826140, 7.35791282457735574780731841299, 7.943443406101927926315376475546
