L(s) = 1 | − 8.08·3-s + 21.9·5-s − 7·7-s + 38.2·9-s + 11·11-s + 16.3·13-s − 177.·15-s − 35.5·17-s + 64.1·19-s + 56.5·21-s + 141.·23-s + 358.·25-s − 91.2·27-s − 268.·29-s + 191.·31-s − 88.8·33-s − 153.·35-s − 101.·37-s − 132.·39-s + 250.·41-s + 523.·43-s + 841.·45-s − 609.·47-s + 49·49-s + 287.·51-s + 0.896·53-s + 241.·55-s + ⋯ |
L(s) = 1 | − 1.55·3-s + 1.96·5-s − 0.377·7-s + 1.41·9-s + 0.301·11-s + 0.349·13-s − 3.05·15-s − 0.506·17-s + 0.774·19-s + 0.587·21-s + 1.28·23-s + 2.86·25-s − 0.650·27-s − 1.72·29-s + 1.10·31-s − 0.468·33-s − 0.743·35-s − 0.452·37-s − 0.543·39-s + 0.954·41-s + 1.85·43-s + 2.78·45-s − 1.89·47-s + 0.142·49-s + 0.788·51-s + 0.00232·53-s + 0.592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.892301929\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892301929\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
good | 3 | \( 1 + 8.08T + 27T^{2} \) |
| 5 | \( 1 - 21.9T + 125T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 250.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 523.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.896T + 1.48e5T^{2} \) |
| 59 | \( 1 + 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 327.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 25.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 200.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 745.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 654.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 13.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 868.T + 9.12e5T^{2} \) |
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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
−9.485372536447798441847349476342, −8.945348448984859884557867371592, −7.28609075673118262469550627980, −6.49960320314823607965694670068, −5.97122083713372909214896409613, −5.38480106674274321644834777653, −4.58209417131968175394741521791, −2.97916955278849454282043775851, −1.70040444022063520283452949374, −0.803206283359199072399585794567,
0.803206283359199072399585794567, 1.70040444022063520283452949374, 2.97916955278849454282043775851, 4.58209417131968175394741521791, 5.38480106674274321644834777653, 5.97122083713372909214896409613, 6.49960320314823607965694670068, 7.28609075673118262469550627980, 8.945348448984859884557867371592, 9.485372536447798441847349476342