L(s) = 1 | + 3-s + 2i·7-s + 9-s − 4i·11-s + 6i·17-s + 4i·19-s + 2i·21-s + 4i·23-s + 27-s + 6i·29-s − 10·31-s − 4i·33-s + 4·37-s + 10·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755i·7-s + 0.333·9-s − 1.20i·11-s + 1.45i·17-s + 0.917i·19-s + 0.436i·21-s + 0.834i·23-s + 0.192·27-s + 1.11i·29-s − 1.79·31-s − 0.696i·33-s + 0.657·37-s + 1.56·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.970374979\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.970374979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 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
−8.934325097315515062710411103075, −8.453941707667243320063162606476, −7.78483334429986629041244145606, −6.83897422448175441910214674755, −5.73841476304653188057490129909, −5.53857771621391824812056342919, −3.97166204253821516020293733532, −3.48541379763484610624814278071, −2.38787358765106114390923444153, −1.36407961934212120403403969598,
0.63354085756665921854310489104, 2.10029165755696696968534195221, 2.86917273189541000697299745877, 4.12175901929702614212381274029, 4.57143196569107446195023826003, 5.59983745828477428832491134402, 6.89076520327156414034535711499, 7.21322568334551017886264946733, 7.906976185720687604870862884506, 8.951106486681739339288857572955