| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4413243698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4413243698\) |
| \(L(1)\) |
\(\approx\) |
\(0.5400249451\) |
| \(L(1)\) |
\(\approx\) |
\(0.5400249451\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−33.86152436170425418701982870885, −32.57269619264754611232473349504, −30.545620946045299761643330146871, −29.966602467693917263919139438815, −28.36648787304449427243069888533, −27.602954185546719470641618338062, −27.13416327234771392576660466512, −25.42092095490063799611210147225, −24.05560065587156226990139008675, −23.35307579450127140854381574674, −21.66216708591945991738451762447, −20.4164355781754680603999999097, −19.05859966143852274240922060246, −18.0793699649316956247127383474, −16.94426102004003420824110155822, −16.00289465879198906094236017551, −14.73731489654954081513575792093, −12.20791887663061997561285230493, −11.45107574999535622633355811796, −10.48461614494641777812922491700, −8.639675813364645526117778754596, −7.44948828207489986312977424102, −6.04706264523221344011253005798, −4.0857712719475633052890036949, −1.28711359688044475280006335894,
1.28711359688044475280006335894, 4.0857712719475633052890036949, 6.04706264523221344011253005798, 7.44948828207489986312977424102, 8.639675813364645526117778754596, 10.48461614494641777812922491700, 11.45107574999535622633355811796, 12.20791887663061997561285230493, 14.73731489654954081513575792093, 16.00289465879198906094236017551, 16.94426102004003420824110155822, 18.0793699649316956247127383474, 19.05859966143852274240922060246, 20.4164355781754680603999999097, 21.66216708591945991738451762447, 23.35307579450127140854381574674, 24.05560065587156226990139008675, 25.42092095490063799611210147225, 27.13416327234771392576660466512, 27.602954185546719470641618338062, 28.36648787304449427243069888533, 29.966602467693917263919139438815, 30.545620946045299761643330146871, 32.57269619264754611232473349504, 33.86152436170425418701982870885
