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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1114509448\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1114509448\) |
\(L(2)\) |
|
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
−9.463614168118140405321683551822, −8.638943834324290413669111600164, −7.86800076596756586325532307822, −7.08138008830288771973417884505, −6.37234392521189484921301045786, −5.69898198017146377196219017218, −4.30096506393193447655690601026, −3.83913046840544296800399662133, −2.96169336834097354255313136891, −1.73740621299698850228428281813,
0.06107455031956434836682323431, 0.43898281667443287308237988768, 2.58463627242419835605285113592, 3.22769802460301738053473178148, 3.88811290613004399552663469790, 5.26864875330780580287984135003, 5.65579289531694526948189081137, 6.99681224094899572631447124918, 7.31779830245384828334347530220, 8.125227650667676466735386688453