L(s) = 1 | − i·3-s + (−2 + i)5-s − i·7-s − 9-s − 4·11-s − 6i·13-s + (1 + 2i)15-s − 2i·17-s − 6·19-s − 21-s + 2i·23-s + (3 − 4i)25-s + i·27-s − 6·29-s − 2·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s − 0.377i·7-s − 0.333·9-s − 1.20·11-s − 1.66i·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s − 1.37·19-s − 0.218·21-s + 0.417i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117582 - 0.498087i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117582 - 0.498087i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.77000054024689993082038004632, −10.26233204862540320742962552216, −8.717159730279984617087983667487, −7.74204907714801789943181887233, −7.43993282822617218684507869671, −6.12496934892994095415489812630, −5.02501258662123917327762216897, −3.61038851945735477649845684264, −2.53221572527109719856893783222, −0.30956948186324356840462234239,
2.27320326542034544972433355508, 3.88532395276813023083743263415, 4.59867280974074603868677014006, 5.73509076737846596235864911372, 6.99613274597008769753838673137, 8.124514529810220712190750848276, 8.806155614910841676191381797347, 9.685442870509197506096491627232, 10.92799676356733792357939437541, 11.32725461725960455700423272904