L(s) = 1 | + 3.24i·5-s − i·7-s − 6.07·11-s + 0.0963·13-s − 3.41i·17-s − 5.89i·19-s + 7.54·23-s − 5.56·25-s + 5.81i·29-s + 1.07i·31-s + 3.24·35-s + 7.82·37-s − 0.614i·41-s − 7.75i·43-s + 0.585·47-s + ⋯ |
L(s) = 1 | + 1.45i·5-s − 0.377i·7-s − 1.83·11-s + 0.0267·13-s − 0.828i·17-s − 1.35i·19-s + 1.57·23-s − 1.11·25-s + 1.07i·29-s + 0.192i·31-s + 0.549·35-s + 1.28·37-s − 0.0958i·41-s − 1.18i·43-s + 0.0854·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548022118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548022118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 3.24iT - 5T^{2} \) |
| 11 | \( 1 + 6.07T + 11T^{2} \) |
| 13 | \( 1 - 0.0963T + 13T^{2} \) |
| 17 | \( 1 + 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 5.89iT - 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 - 5.81iT - 29T^{2} \) |
| 31 | \( 1 - 1.07iT - 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 + 0.614iT - 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 - 0.585T + 47T^{2} \) |
| 53 | \( 1 - 7.39iT - 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 9.19T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 3.01iT - 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.87741362493891456817013076962, −7.32239841020660246860016097628, −6.97403847626819071813965293664, −6.16645039240651936337803750793, −5.13841283104284438584973739379, −4.76872023318225199712368887438, −3.42664841136651878656592882771, −2.84064677557649676571379900505, −2.37292424376077574886755864900, −0.70143804855573540336420300465,
0.59000010012690945533322683630, 1.66746313885331905129269781127, 2.58790426217531541825395658355, 3.56568024367050850623630559081, 4.62170401685743578915797683343, 5.02388510262064869326466260908, 5.75985837076499415046225789550, 6.32881846679603098267295586294, 7.71121968376370051235180430674, 7.981312263550060297030100655935