L(s) = 1 | + (−1.28 + 0.599i)2-s + (1.66 − 0.468i)3-s + (1.28 − 1.53i)4-s + (−1.85 + 1.59i)6-s − 0.936·7-s + (−0.719 + 2.73i)8-s + (2.56 − 1.56i)9-s + 4.27·11-s + (1.41 − 3.16i)12-s + 3.12i·13-s + (1.19 − 0.561i)14-s + (−0.719 − 3.93i)16-s + 2·17-s + (−2.34 + 3.53i)18-s − 4.27i·19-s + ⋯ |
L(s) = 1 | + (−0.905 + 0.424i)2-s + (0.962 − 0.270i)3-s + (0.640 − 0.768i)4-s + (−0.757 + 0.653i)6-s − 0.353·7-s + (−0.254 + 0.967i)8-s + (0.853 − 0.520i)9-s + 1.28·11-s + (0.408 − 0.912i)12-s + 0.866i·13-s + (0.320 − 0.150i)14-s + (−0.179 − 0.983i)16-s + 0.485·17-s + (−0.552 + 0.833i)18-s − 0.979i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23594 + 0.0261890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23594 + 0.0261890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 - 0.599i)T \) |
| 3 | \( 1 + (-1.66 + 0.468i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.936T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 - 3.12iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.27iT - 19T^{2} \) |
| 23 | \( 1 + 7.60iT - 23T^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 - 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 0.936iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 5.20T + 67T^{2} \) |
| 71 | \( 1 + 6.67T + 71T^{2} \) |
| 73 | \( 1 + 8.24iT - 73T^{2} \) |
| 79 | \( 1 - 9.06iT - 79T^{2} \) |
| 83 | \( 1 - 4.68iT - 83T^{2} \) |
| 89 | \( 1 - 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−11.68570208316590241962551077961, −10.53793631969664976425419091880, −9.417523879667705201678344065927, −9.023814990833136627649194588733, −8.073977682338736876866331161819, −6.85192111908180373358237743062, −6.47408190478846246138257891776, −4.54781125452816927495028166260, −2.95419845830652817350369686750, −1.44329614820325290940510659740,
1.60059385969909256197610543930, 3.16828293567293749481697978838, 3.96792531876273576665322465797, 5.99556006745639994814437472833, 7.36225566979005029315316518748, 8.024202217496491676259763224758, 9.123471852429395801357601514954, 9.683058502886444247891084142330, 10.49081153665844061965775768696, 11.65375931800909127968101759253