L(s) = 1 | + (0.207 + 0.158i)3-s + (−0.241 − 0.900i)9-s + (−0.0675 + 0.513i)11-s + (−0.128 − 1.95i)17-s + (−0.923 − 1.38i)19-s + (−0.608 − 0.793i)25-s + (0.192 − 0.465i)27-s + (−0.0955 + 0.0955i)33-s + (1.05 − 0.357i)41-s + (−0.410 + 1.53i)43-s + (0.991 + 0.130i)49-s + (0.284 − 0.425i)51-s + (0.0283 − 0.433i)57-s + (0.257 + 0.293i)59-s + (1.57 − 1.05i)67-s + ⋯ |
L(s) = 1 | + (0.207 + 0.158i)3-s + (−0.241 − 0.900i)9-s + (−0.0675 + 0.513i)11-s + (−0.128 − 1.95i)17-s + (−0.923 − 1.38i)19-s + (−0.608 − 0.793i)25-s + (0.192 − 0.465i)27-s + (−0.0955 + 0.0955i)33-s + (1.05 − 0.357i)41-s + (−0.410 + 1.53i)43-s + (0.991 + 0.130i)49-s + (0.284 − 0.425i)51-s + (0.0283 − 0.433i)57-s + (0.257 + 0.293i)59-s + (1.57 − 1.05i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.100595983\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100595983\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 + (0.793 + 0.608i)T \) |
good | 3 | \( 1 + (-0.207 - 0.158i)T + (0.258 + 0.965i)T^{2} \) |
| 5 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 7 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 11 | \( 1 + (0.0675 - 0.513i)T + (-0.965 - 0.258i)T^{2} \) |
| 13 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 17 | \( 1 + (0.128 + 1.95i)T + (-0.991 + 0.130i)T^{2} \) |
| 19 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 23 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 29 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 31 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 37 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 0.357i)T + (0.793 - 0.608i)T^{2} \) |
| 43 | \( 1 + (0.410 - 1.53i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 59 | \( 1 + (-0.257 - 0.293i)T + (-0.130 + 0.991i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 1.05i)T + (0.382 - 0.923i)T^{2} \) |
| 71 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 73 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (1.50 - 0.0983i)T + (0.991 - 0.130i)T^{2} \) |
| 89 | \( 1 + (1.12 - 0.465i)T + (0.707 - 0.707i)T^{2} \) |
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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
−8.841132662396466021991819439331, −8.051014113988557240143178145862, −7.07588681225569484351274685595, −6.65715725576688099690521645807, −5.67971229344761344918188463177, −4.72232347847113313240748201334, −4.14369019923536391000171761255, −2.94914985468286081625294431608, −2.34925109558118133221651070436, −0.64576724806231953065861426128,
1.59984221475089465836789092206, 2.34052816441802697697617696023, 3.62754927260318928381352669345, 4.14588438121690093189640279078, 5.44501787359434701888215029539, 5.86974362211968950700419100202, 6.76737189081060869401591275106, 7.75966656881997772296137487313, 8.284818384137713390637196575219, 8.736347362350989212313062800588