L(s) = 1 | − 2-s + 3-s + 4-s − 4.19·5-s − 6-s − 2.48·7-s − 8-s + 9-s + 4.19·10-s − 3.45·11-s + 12-s − 13-s + 2.48·14-s − 4.19·15-s + 16-s + 4.15·17-s − 18-s − 1.49·19-s − 4.19·20-s − 2.48·21-s + 3.45·22-s − 1.16·23-s − 24-s + 12.5·25-s + 26-s + 27-s − 2.48·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.87·5-s − 0.408·6-s − 0.939·7-s − 0.353·8-s + 0.333·9-s + 1.32·10-s − 1.04·11-s + 0.288·12-s − 0.277·13-s + 0.664·14-s − 1.08·15-s + 0.250·16-s + 1.00·17-s − 0.235·18-s − 0.342·19-s − 0.937·20-s − 0.542·21-s + 0.736·22-s − 0.243·23-s − 0.204·24-s + 2.51·25-s + 0.196·26-s + 0.192·27-s − 0.469·28-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 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 0.938T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 0.896T + 71T^{2} \) |
| 73 | \( 1 + 0.857T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 1.28T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−7.70009718417106011405311805058, −7.01048979120295781013645337034, −6.50492793813170097873875387268, −5.32394574667794908464289732458, −4.51337731238503125788857407997, −3.61481026255909027488106119830, −3.14645628790322518453465624400, −2.42868345163478092382722836009, −0.897430604946334436060671102123, 0,
0.897430604946334436060671102123, 2.42868345163478092382722836009, 3.14645628790322518453465624400, 3.61481026255909027488106119830, 4.51337731238503125788857407997, 5.32394574667794908464289732458, 6.50492793813170097873875387268, 7.01048979120295781013645337034, 7.70009718417106011405311805058