| L(s) = 1 | − 4i·5-s + 6.92i·7-s − 6.92·11-s + 18·17-s − 20.7·19-s − 41.5i·23-s + 9·25-s − 4i·29-s + 48.4i·31-s + 27.7·35-s − 72i·37-s − 18·41-s + 62.3·43-s − 41.5i·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | − 0.800i·5-s + 0.989i·7-s − 0.629·11-s + 1.05·17-s − 1.09·19-s − 1.80i·23-s + 0.359·25-s − 0.137i·29-s + 1.56i·31-s + 0.791·35-s − 1.94i·37-s − 0.439·41-s + 1.45·43-s − 0.884i·47-s + 0.0204·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.364972149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364972149\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4iT - 25T^{2} \) |
| 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 + 41.5iT - 529T^{2} \) |
| 29 | \( 1 + 4iT - 841T^{2} \) |
| 31 | \( 1 - 48.4iT - 961T^{2} \) |
| 37 | \( 1 + 72iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 72iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 20.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 131.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 126T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110T + 9.40e3T^{2} \) |
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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
−9.133873874771819728929222902233, −8.674550107585541191699416212464, −7.992164910790631451101724566347, −6.86422012190438359078361645025, −5.84985096964436186265942254322, −5.17344111271199071748647454401, −4.31167353552181212593726436827, −2.96244552248747365841215365970, −1.96490530206499529636561366592, −0.44067603246664855991975339588,
1.19489227647182569626389076944, 2.66505200497999106628863372588, 3.57673556573545803613706538956, 4.52605682725075321197939644386, 5.68859687183778914514713600335, 6.50491828713032080648408296047, 7.57770992604801938194738302172, 7.75813375415866529586295141916, 9.096681348721492620924325311785, 10.03445322909831612638309236215
