L(s) = 1 | + 3-s − 5-s + 2.73·7-s − 2·9-s − 2.73·11-s − 1.73·13-s − 15-s + 5.46·17-s − 2.73·19-s + 2.73·21-s − 23-s + 25-s − 5·27-s − 5.92·29-s − 0.267·31-s − 2.73·33-s − 2.73·35-s − 2.73·37-s − 1.73·39-s + 2.46·41-s − 6.73·43-s + 2·45-s − 5.92·47-s + 0.464·49-s + 5.46·51-s − 0.535·53-s + 2.73·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.03·7-s − 0.666·9-s − 0.823·11-s − 0.480·13-s − 0.258·15-s + 1.32·17-s − 0.626·19-s + 0.596·21-s − 0.208·23-s + 0.200·25-s − 0.962·27-s − 1.10·29-s − 0.0481·31-s − 0.475·33-s − 0.461·35-s − 0.449·37-s − 0.277·39-s + 0.384·41-s − 1.02·43-s + 0.298·45-s − 0.864·47-s + 0.0663·49-s + 0.765·51-s − 0.0736·53-s + 0.368·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 + 0.267T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.66T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−8.140229103645602509662231301235, −7.70892620601337013979370877007, −6.90722462804769676410552942144, −5.62704843522454210109677284487, −5.25384610666469510982419332807, −4.26050578600970031803516595451, −3.37666690791159365347927213654, −2.55794171787847280934607824248, −1.60614036440783641206313733085, 0,
1.60614036440783641206313733085, 2.55794171787847280934607824248, 3.37666690791159365347927213654, 4.26050578600970031803516595451, 5.25384610666469510982419332807, 5.62704843522454210109677284487, 6.90722462804769676410552942144, 7.70892620601337013979370877007, 8.140229103645602509662231301235