| L(s) = 1 | + 1.59e6·3-s − 6.93e8·5-s − 3.81e11·7-s + 2.54e12·9-s − 1.26e14·11-s − 1.31e15·13-s − 1.10e15·15-s − 3.53e16·17-s − 8.65e16·19-s − 6.07e17·21-s − 5.10e17·23-s − 6.96e18·25-s + 4.05e18·27-s − 2.74e19·29-s + 9.77e19·31-s − 2.01e20·33-s + 2.64e20·35-s − 1.80e21·37-s − 2.09e21·39-s − 5.17e21·41-s − 8.74e21·43-s − 1.76e21·45-s − 2.51e22·47-s + 7.94e22·49-s − 5.63e22·51-s + 2.11e23·53-s + 8.76e22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.254·5-s − 1.48·7-s + 0.333·9-s − 1.10·11-s − 1.20·13-s − 0.146·15-s − 0.865·17-s − 0.471·19-s − 0.858·21-s − 0.211·23-s − 0.935·25-s + 0.192·27-s − 0.497·29-s + 0.719·31-s − 0.637·33-s + 0.377·35-s − 1.21·37-s − 0.695·39-s − 0.874·41-s − 0.776·43-s − 0.0847·45-s − 0.672·47-s + 1.20·49-s − 0.499·51-s + 1.11·53-s + 0.280·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(0.2264163283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2264163283\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.59e6T \) |
| good | 5 | \( 1 + 6.93e8T + 7.45e18T^{2} \) |
| 7 | \( 1 + 3.81e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 1.26e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.31e15T + 1.19e30T^{2} \) |
| 17 | \( 1 + 3.53e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 8.65e16T + 3.36e34T^{2} \) |
| 23 | \( 1 + 5.10e17T + 5.84e36T^{2} \) |
| 29 | \( 1 + 2.74e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 9.77e19T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.80e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 5.17e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 8.74e21T + 1.26e44T^{2} \) |
| 47 | \( 1 + 2.51e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.11e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 7.00e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 1.08e23T + 1.59e48T^{2} \) |
| 67 | \( 1 + 2.46e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 1.29e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 7.46e24T + 2.04e50T^{2} \) |
| 79 | \( 1 - 4.94e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 5.54e25T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.32e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 1.06e27T + 4.39e53T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36287109698087618041936899701, −9.710660307821038873161669788459, −8.548475112804901929591898438491, −7.41519793902410219430235538710, −6.49966105829736994258851678019, −5.12248240304278115737188972077, −3.86213221990743207008949767094, −2.86339390631787131268779890718, −2.07279249593565891987722235362, −0.17316656492519184755439460596,
0.17316656492519184755439460596, 2.07279249593565891987722235362, 2.86339390631787131268779890718, 3.86213221990743207008949767094, 5.12248240304278115737188972077, 6.49966105829736994258851678019, 7.41519793902410219430235538710, 8.548475112804901929591898438491, 9.710660307821038873161669788459, 10.36287109698087618041936899701
