L(s) = 1 | − 2-s + 4-s − 5-s − 3.12·7-s − 8-s + 10-s + 4·11-s + 0.438·13-s + 3.12·14-s + 16-s − 17-s + 1.56·19-s − 20-s − 4·22-s − 3.12·23-s + 25-s − 0.438·26-s − 3.12·28-s − 6.68·29-s − 2.43·31-s − 32-s + 34-s + 3.12·35-s + 1.12·37-s − 1.56·38-s + 40-s + 12.2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.18·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.121·13-s + 0.834·14-s + 0.250·16-s − 0.242·17-s + 0.358·19-s − 0.223·20-s − 0.852·22-s − 0.651·23-s + 0.200·25-s − 0.0859·26-s − 0.590·28-s − 1.24·29-s − 0.437·31-s − 0.176·32-s + 0.171·34-s + 0.527·35-s + 0.184·37-s − 0.253·38-s + 0.158·40-s + 1.91·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9058673414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9058673414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.468100006641804358571245512954, −8.844605072055641474636607187435, −7.87128192079276210509186977477, −7.07188927517538384852063775957, −6.41242106312465046065176472447, −5.61820900506025136928212739955, −4.08471565506594536747870513080, −3.48409148095670405928575117109, −2.22637232009706507306178055755, −0.73843247365846022748846151101,
0.73843247365846022748846151101, 2.22637232009706507306178055755, 3.48409148095670405928575117109, 4.08471565506594536747870513080, 5.61820900506025136928212739955, 6.41242106312465046065176472447, 7.07188927517538384852063775957, 7.87128192079276210509186977477, 8.844605072055641474636607187435, 9.468100006641804358571245512954