L(s) = 1 | + (−0.559 + 0.418i)3-s + (−0.144 + 0.490i)9-s + (−0.909 + 0.584i)11-s + (−0.989 + 0.857i)17-s + (0.373 − 0.203i)19-s + (−0.415 − 0.909i)25-s + (−0.369 − 0.989i)27-s + (0.264 − 0.708i)33-s + (−0.847 + 1.13i)41-s + (−1.94 − 0.424i)43-s + (−0.909 + 0.415i)49-s + (0.194 − 0.894i)51-s + (−0.123 + 0.270i)57-s + (0.114 + 0.0855i)59-s + (−0.0801 + 0.557i)67-s + ⋯ |
L(s) = 1 | + (−0.559 + 0.418i)3-s + (−0.144 + 0.490i)9-s + (−0.909 + 0.584i)11-s + (−0.989 + 0.857i)17-s + (0.373 − 0.203i)19-s + (−0.415 − 0.909i)25-s + (−0.369 − 0.989i)27-s + (0.264 − 0.708i)33-s + (−0.847 + 1.13i)41-s + (−1.94 − 0.424i)43-s + (−0.909 + 0.415i)49-s + (0.194 − 0.894i)51-s + (−0.123 + 0.270i)57-s + (0.114 + 0.0855i)59-s + (−0.0801 + 0.557i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3554645057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3554645057\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
good | 3 | \( 1 + (0.559 - 0.418i)T + (0.281 - 0.959i)T^{2} \) |
| 5 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 11 | \( 1 + (0.909 - 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 17 | \( 1 + (0.989 - 0.857i)T + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.373 + 0.203i)T + (0.540 - 0.841i)T^{2} \) |
| 23 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 29 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 31 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.847 - 1.13i)T + (-0.281 - 0.959i)T^{2} \) |
| 43 | \( 1 + (1.94 + 0.424i)T + (0.909 + 0.415i)T^{2} \) |
| 47 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 53 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.114 - 0.0855i)T + (0.281 + 0.959i)T^{2} \) |
| 61 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 67 | \( 1 + (0.0801 - 0.557i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (-1.74 + 0.512i)T + (0.841 - 0.540i)T^{2} \) |
| 79 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (0.114 - 1.59i)T + (-0.989 - 0.142i)T^{2} \) |
| 97 | \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \) |
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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
−9.533694356713479836486898276742, −8.283074668853291080272536932056, −8.094721772567565237813422203002, −6.93159479932943104633519269796, −6.27302030902379510631334996649, −5.29668819272796911414073324716, −4.80382223894381444972031455617, −3.97759679517490221102619509585, −2.74544438683706450935709681231, −1.83788071266908655277063295775,
0.22604737095680803651617071041, 1.69128425197575870673691722950, 2.91972000082685611214866455579, 3.70498001887695097604530629019, 5.01562417270557909534876648820, 5.45045797524849348104007187253, 6.42161556123362518681118549702, 6.96405558781418084296655673094, 7.80185301116003868450455823438, 8.589017867820179538414855778171