L(s) = 1 | − 2-s + 3-s + 4-s − 1.61·5-s − 6-s − 8-s + 9-s + 1.61·10-s − 4·11-s + 12-s − 0.618·13-s − 1.61·15-s + 16-s + 7.09·17-s − 18-s − 1.61·20-s + 4·22-s − 4·23-s − 24-s − 2.38·25-s + 0.618·26-s + 27-s − 2.14·29-s + 1.61·30-s + 10.4·31-s − 32-s − 4·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.723·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.511·10-s − 1.20·11-s + 0.288·12-s − 0.171·13-s − 0.417·15-s + 0.250·16-s + 1.71·17-s − 0.235·18-s − 0.361·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 0.476·25-s + 0.121·26-s + 0.192·27-s − 0.398·29-s + 0.295·30-s + 1.88·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.14T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 0.854T + 37T^{2} \) |
| 41 | \( 1 + 0.854T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6.38T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.319919433686698421838341544832, −8.024977885173213731235973985050, −7.58423129413612126555166449990, −6.53844986343870282741077518178, −5.55221809243616450887418830939, −4.58247138454900292960903380328, −3.43602963638392449961245525625, −2.78222236276994469773121213760, −1.52329535296591515958877936932, 0,
1.52329535296591515958877936932, 2.78222236276994469773121213760, 3.43602963638392449961245525625, 4.58247138454900292960903380328, 5.55221809243616450887418830939, 6.53844986343870282741077518178, 7.58423129413612126555166449990, 8.024977885173213731235973985050, 8.319919433686698421838341544832