L(s) = 1 | + (1.74 − 0.512i)3-s + (−0.841 − 0.540i)5-s + (0.281 + 1.95i)7-s + (1.94 − 1.24i)9-s + (−1.74 − 0.512i)15-s + (1.49 + 3.27i)21-s + (0.909 − 0.415i)23-s + (0.415 + 0.909i)25-s + (1.56 − 1.80i)27-s + (−1.10 − 1.27i)29-s + (0.822 − 1.80i)35-s + (0.239 + 0.153i)41-s + (−1.45 + 0.425i)43-s − 2.30·45-s + 1.08·47-s + ⋯ |
L(s) = 1 | + (1.74 − 0.512i)3-s + (−0.841 − 0.540i)5-s + (0.281 + 1.95i)7-s + (1.94 − 1.24i)9-s + (−1.74 − 0.512i)15-s + (1.49 + 3.27i)21-s + (0.909 − 0.415i)23-s + (0.415 + 0.909i)25-s + (1.56 − 1.80i)27-s + (−1.10 − 1.27i)29-s + (0.822 − 1.80i)35-s + (0.239 + 0.153i)41-s + (−1.45 + 0.425i)43-s − 2.30·45-s + 1.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.866859409\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.866859409\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.909 + 0.415i)T \) |
good | 3 | \( 1 + (-1.74 + 0.512i)T + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.281 - 1.95i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - 1.08T + T^{2} \) |
| 53 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.474 + 0.304i)T + (0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
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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
−9.161615616600336514323499768446, −8.576212770708775020796030254972, −8.081428314044007445897482199056, −7.43921215541309699387254186945, −6.37303754905028210369765358821, −5.28500172497135730065884879013, −4.31228136843611206717902299671, −3.26301314462408357534215953079, −2.54752656609172453208102995809, −1.64059223636043672716229268219,
1.48591092466164564149388624914, 2.93532223533742925427839506041, 3.63725824264034833447181907532, 4.10863810968350020015461460645, 4.96284092967055379055375293543, 6.88295233931080816453108398447, 7.31441074379581097551158192448, 7.82435363833509327427360210778, 8.605919990283271892872491157246, 9.388133441136257108482546120307