| L(s) = 1 | + 0.0296·2-s − 3-s − 1.99·4-s + 3.07·5-s − 0.0296·6-s − 7-s − 0.118·8-s + 9-s + 0.0913·10-s − 4.20·11-s + 1.99·12-s − 6.43·13-s − 0.0296·14-s − 3.07·15-s + 3.99·16-s − 0.978·17-s + 0.0296·18-s + 0.452·19-s − 6.15·20-s + 21-s − 0.124·22-s + 1.87·23-s + 0.118·24-s + 4.48·25-s − 0.190·26-s − 27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | + 0.0209·2-s − 0.577·3-s − 0.999·4-s + 1.37·5-s − 0.0121·6-s − 0.377·7-s − 0.0419·8-s + 0.333·9-s + 0.0288·10-s − 1.26·11-s + 0.577·12-s − 1.78·13-s − 0.00792·14-s − 0.795·15-s + 0.998·16-s − 0.237·17-s + 0.00699·18-s + 0.103·19-s − 1.37·20-s + 0.218·21-s − 0.0266·22-s + 0.391·23-s + 0.0242·24-s + 0.896·25-s − 0.0374·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.000040440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000040440\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0296T + 2T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 + 0.978T + 17T^{2} \) |
| 19 | \( 1 - 0.452T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 - 4.11T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 0.989T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 - 9.02T + 79T^{2} \) |
| 83 | \( 1 + 5.68T + 83T^{2} \) |
| 89 | \( 1 - 2.01T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.108642898165902457508221854895, −8.110494920039174504989109589914, −7.29330302509586434973378647413, −6.39991400907715229389802680570, −5.52321555787388106201273825325, −5.10949882534811626996101075040, −4.43499761356964604601261555508, −2.96240661689041571512463288256, −2.18245555956808545319297098120, −0.62523203638634136712268818242,
0.62523203638634136712268818242, 2.18245555956808545319297098120, 2.96240661689041571512463288256, 4.43499761356964604601261555508, 5.10949882534811626996101075040, 5.52321555787388106201273825325, 6.39991400907715229389802680570, 7.29330302509586434973378647413, 8.110494920039174504989109589914, 9.108642898165902457508221854895
