L(s) = 1 | + (2.44 − 2.44i)5-s − 1.41i·7-s + (3.46 − 3.46i)11-s + (1 + i)13-s − 4.89·17-s + (−4.24 − 4.24i)19-s − 6.92i·23-s − 6.99i·25-s + (2.44 + 2.44i)29-s − 1.41·31-s + (−3.46 − 3.46i)35-s + (−7 + 7i)37-s + 4.89i·41-s + (4.24 − 4.24i)43-s + 6.92·47-s + ⋯ |
L(s) = 1 | + (1.09 − 1.09i)5-s − 0.534i·7-s + (1.04 − 1.04i)11-s + (0.277 + 0.277i)13-s − 1.18·17-s + (−0.973 − 0.973i)19-s − 1.44i·23-s − 1.39i·25-s + (0.454 + 0.454i)29-s − 0.254·31-s + (−0.585 − 0.585i)35-s + (−1.15 + 1.15i)37-s + 0.765i·41-s + (0.646 − 0.646i)43-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.027911720\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.027911720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (-3.46 + 3.46i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + (4.24 + 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + (-2.44 - 2.44i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + (2.44 - 2.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.92 - 6.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5 - 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.65 + 5.65i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.872191775170555175028541521441, −8.425093486286266018131868464723, −6.92487084216159527147771724067, −6.44314480628535337461076141504, −5.70002976251089425435403962026, −4.60940239039648567022927238347, −4.18984415617219283337006036520, −2.76639957035120494206027836990, −1.65503316475508188382807631492, −0.67127771469963391877344794094,
1.79994624052672298707906261336, 2.23337519031849201929505336785, 3.47806485843518666826652473220, 4.34929905043491066872926324748, 5.59278466989751962633776800909, 6.13762570649127747420234551178, 6.83323401835243597105502970590, 7.49172012230195046985076496831, 8.707967911318140472618559526871, 9.297435532693924856838095127173