| L(s) = 1 | − 3.37·3-s + 14.4·5-s − 7.73·7-s − 15.6·9-s + 22.3·11-s + 16.3·13-s − 48.6·15-s − 116.·17-s + 25.6·19-s + 26.0·21-s − 161.·23-s + 83.4·25-s + 143.·27-s + 191.·29-s − 100.·31-s − 75.4·33-s − 111.·35-s − 37·37-s − 55.1·39-s + 318.·41-s − 205.·43-s − 225.·45-s − 60.6·47-s − 283.·49-s + 392.·51-s − 726.·53-s + 323.·55-s + ⋯ |
| L(s) = 1 | − 0.648·3-s + 1.29·5-s − 0.417·7-s − 0.578·9-s + 0.613·11-s + 0.348·13-s − 0.838·15-s − 1.66·17-s + 0.309·19-s + 0.270·21-s − 1.46·23-s + 0.667·25-s + 1.02·27-s + 1.22·29-s − 0.583·31-s − 0.398·33-s − 0.539·35-s − 0.164·37-s − 0.226·39-s + 1.21·41-s − 0.728·43-s − 0.747·45-s − 0.188·47-s − 0.825·49-s + 1.07·51-s − 1.88·53-s + 0.792·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 + 3.37T + 27T^{2} \) |
| 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 + 7.73T + 343T^{2} \) |
| 11 | \( 1 - 22.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 77.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 586.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 880.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 277.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 672.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 280.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
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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
−9.815201946532941906187637496688, −9.151335345497156010750498508077, −8.238009824892990038416628008584, −6.62200581827996941156923397868, −6.28265321204841955682471311948, −5.43303592401831655380470427966, −4.28346118469619267571868500361, −2.79086553256750125344511698199, −1.62189891312527077082284932171, 0,
1.62189891312527077082284932171, 2.79086553256750125344511698199, 4.28346118469619267571868500361, 5.43303592401831655380470427966, 6.28265321204841955682471311948, 6.62200581827996941156923397868, 8.238009824892990038416628008584, 9.151335345497156010750498508077, 9.815201946532941906187637496688
