L(s) = 1 | − 2-s + 3-s + 4-s + 1.50·5-s − 6-s + 0.210·7-s − 8-s + 9-s − 1.50·10-s − 5.97·11-s + 12-s − 13-s − 0.210·14-s + 1.50·15-s + 16-s + 5.33·17-s − 18-s − 3.08·19-s + 1.50·20-s + 0.210·21-s + 5.97·22-s − 1.61·23-s − 24-s − 2.74·25-s + 26-s + 27-s + 0.210·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.672·5-s − 0.408·6-s + 0.0796·7-s − 0.353·8-s + 0.333·9-s − 0.475·10-s − 1.80·11-s + 0.288·12-s − 0.277·13-s − 0.0563·14-s + 0.387·15-s + 0.250·16-s + 1.29·17-s − 0.235·18-s − 0.708·19-s + 0.336·20-s + 0.0459·21-s + 1.27·22-s − 0.337·23-s − 0.204·24-s − 0.548·25-s + 0.196·26-s + 0.192·27-s + 0.0398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 - 0.210T + 7T^{2} \) |
| 11 | \( 1 + 5.97T + 11T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 4.95T + 71T^{2} \) |
| 73 | \( 1 - 9.33T + 73T^{2} \) |
| 79 | \( 1 + 0.938T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.82488384432400733742897104381, −6.96753836233853288170782312727, −6.16462584929145888667738538648, −5.43386005449570865268997634941, −4.83848866021862821752842061329, −3.67338291434436487494216619172, −2.77057948691122066060088236655, −2.29023514978175452958804198951, −1.34642218403042467255483042642, 0,
1.34642218403042467255483042642, 2.29023514978175452958804198951, 2.77057948691122066060088236655, 3.67338291434436487494216619172, 4.83848866021862821752842061329, 5.43386005449570865268997634941, 6.16462584929145888667738538648, 6.96753836233853288170782312727, 7.82488384432400733742897104381