| L(s) = 1 | + 4.87·5-s − 11.7·7-s + 9.75·11-s − 22.6i·13-s + 22.1i·17-s − 17.7i·19-s − 14.1i·23-s − 1.20·25-s + 20.0·29-s − 39.7·31-s − 57.5·35-s + 2.40i·37-s + 64.3i·41-s − 3.19i·43-s + 41.8i·47-s + ⋯ |
| L(s) = 1 | + 0.975·5-s − 1.68·7-s + 0.886·11-s − 1.74i·13-s + 1.30i·17-s − 0.936i·19-s − 0.614i·23-s − 0.0480·25-s + 0.689·29-s − 1.28·31-s − 1.64·35-s + 0.0649i·37-s + 1.56i·41-s − 0.0742i·43-s + 0.890i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4489229524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4489229524\) |
| \(L(2)\) |
|
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 | \( 1 - 4.87T + 25T^{2} \) |
| 7 | \( 1 + 11.7T + 49T^{2} \) |
| 11 | \( 1 - 9.75T + 121T^{2} \) |
| 13 | \( 1 + 22.6iT - 169T^{2} \) |
| 17 | \( 1 - 22.1iT - 289T^{2} \) |
| 19 | \( 1 + 17.7iT - 361T^{2} \) |
| 23 | \( 1 + 14.1iT - 529T^{2} \) |
| 29 | \( 1 - 20.0T + 841T^{2} \) |
| 31 | \( 1 + 39.7T + 961T^{2} \) |
| 37 | \( 1 - 2.40iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 64.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.19iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 55.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 111.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 10.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 34.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 151.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 61.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 72.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.3T + 9.40e3T^{2} \) |
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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
−8.691367471725369863033381358539, −7.64052582045562376002961685243, −6.64906993113958586851749284262, −6.11894370513272715274791096529, −5.63016635782792518253313004491, −4.36014699081053079894194617166, −3.29912241431356359440585061080, −2.72560067610033576094344199780, −1.36758239930099737452851645717, −0.10470774765350301249188376254,
1.46335246855625257905858826657, 2.41526244783619506613706232542, 3.50431835911368687507204001865, 4.20579269435621478414566870131, 5.50868638787567658833155396502, 6.10916492523305654779309573417, 6.86781015958356837683778355539, 7.26213471998692405181914669787, 8.944097965191101160486251951966, 9.157321751908176536139093926026
