L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.453 + 0.891i)5-s + (−0.309 + 0.951i)8-s + (−0.707 − 0.707i)9-s + (−0.156 + 0.987i)10-s + (1.10 − 0.678i)13-s + (−0.809 + 0.587i)16-s + (−1.04 + 0.0819i)17-s + (−0.156 − 0.987i)18-s + (−0.707 + 0.707i)20-s + (−0.587 + 0.809i)25-s + (1.29 + 0.101i)26-s + (−1.20 − 1.40i)29-s − 32-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.453 + 0.891i)5-s + (−0.309 + 0.951i)8-s + (−0.707 − 0.707i)9-s + (−0.156 + 0.987i)10-s + (1.10 − 0.678i)13-s + (−0.809 + 0.587i)16-s + (−1.04 + 0.0819i)17-s + (−0.156 − 0.987i)18-s + (−0.707 + 0.707i)20-s + (−0.587 + 0.809i)25-s + (1.29 + 0.101i)26-s + (−1.20 − 1.40i)29-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.534073992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.534073992\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.453 - 0.891i)T \) |
| 41 | \( 1 + (-0.587 + 0.809i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.891 + 0.453i)T^{2} \) |
| 11 | \( 1 + (0.987 - 0.156i)T^{2} \) |
| 13 | \( 1 + (-1.10 + 0.678i)T + (0.453 - 0.891i)T^{2} \) |
| 17 | \( 1 + (1.04 - 0.0819i)T + (0.987 - 0.156i)T^{2} \) |
| 19 | \( 1 + (-0.891 + 0.453i)T^{2} \) |
| 23 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (1.20 + 1.40i)T + (-0.156 + 0.987i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.69 - 0.863i)T + (0.587 + 0.809i)T^{2} \) |
| 43 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 47 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 53 | \( 1 + (-0.0366 + 0.465i)T + (-0.987 - 0.156i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 67 | \( 1 + (-0.156 + 0.987i)T^{2} \) |
| 71 | \( 1 + (-0.987 + 0.156i)T^{2} \) |
| 73 | \( 1 - 0.907iT - T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.399 - 0.652i)T + (-0.453 - 0.891i)T^{2} \) |
| 97 | \( 1 + (-0.101 - 0.119i)T + (-0.156 + 0.987i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06871051715106773773214789948, −9.716797535926306986761855040750, −8.779574665424826288509569661426, −7.930979738963985328322182335153, −6.94379474824238067073938084323, −6.04979221453964543294849095266, −5.76763208745468996892240140440, −4.22642887388134816010937102739, −3.33330260766177223983776955659, −2.37703782299849093310906873650,
1.51390013658146603808511292252, 2.58629967143707761821798279230, 3.98730510181682682733094197788, 4.79114019441087826219153977626, 5.72167695094776149557329774032, 6.35736631730352157186125187953, 7.69177748775900247362832999245, 9.019187847906700389859326668807, 9.181753481273746722837377263337, 10.60070238345724676974551854223