L(s) = 1 | + (−7.38 + 3.07i)2-s + (45.0 − 45.4i)4-s + 70.4i·5-s − 87.7i·7-s + (−193. + 474. i)8-s + (−216. − 520. i)10-s + 407.·11-s − 1.89e3i·13-s + (270. + 648. i)14-s + (−33.3 − 4.09e3i)16-s + 2.51e3·17-s + 437.·19-s + (3.20e3 + 3.17e3i)20-s + (−3.00e3 + 1.25e3i)22-s + 1.40e4i·23-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.384i)2-s + (0.704 − 0.709i)4-s + 0.563i·5-s − 0.255i·7-s + (−0.377 + 0.926i)8-s + (−0.216 − 0.520i)10-s + 0.305·11-s − 0.861i·13-s + (0.0984 + 0.236i)14-s + (−0.00813 − 0.999i)16-s + 0.511·17-s + 0.0638·19-s + (0.400 + 0.396i)20-s + (−0.282 + 0.117i)22-s + 1.15i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.944437 + 0.635226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.944437 + 0.635226i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (7.38 - 3.07i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 70.4iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 87.7iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 407.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.89e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 2.51e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 437.T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.40e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.15e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.84e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 5.50e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.57e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 8.99e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 4.40e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.74e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.75e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.81e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.80e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.01e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 7.40e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.17e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.05e5T + 8.32e11T^{2} \) |
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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
−13.95043431187818275283107188561, −12.33467438901469405126694899188, −11.00875448888083673107011171161, −10.24497195534568820046848272682, −9.030749517405967394049439380940, −7.74582052298044206490325418228, −6.76867949808081292864723882454, −5.39408778891404339972163834213, −3.08865901187157745589391218820, −1.16315793031807131012515106333,
0.71911722945033265403846838973, 2.32069476467525288971736118632, 4.18972321896254706485498246703, 6.17634532478694084901913365089, 7.60629540363273769526037238695, 8.808980060983517084225112332111, 9.607500666365840392136378196717, 10.94177555665323085599023988103, 11.99549722306521157778612561732, 12.82946209071871718038244906314