L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 4·13-s + 15-s + 6·17-s − 19-s + 2·21-s − 6·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 2·35-s − 4·37-s + 4·39-s − 6·41-s + 10·43-s − 45-s − 6·47-s − 3·49-s − 6·51-s + 12·53-s + 57-s − 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.338·35-s − 0.657·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s + 0.132·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8075951929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8075951929\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.145678646362832718573957535238, −7.55609912020355969196380780205, −6.89994642339545498102283322245, −6.11826238957396227785655637585, −5.42832276623074742721237420550, −4.66672123451459212142344880481, −3.75530481705680362889110902094, −3.04134190023796954341741481673, −1.87930770721828041988463893093, −0.50095712833359203238640449289,
0.50095712833359203238640449289, 1.87930770721828041988463893093, 3.04134190023796954341741481673, 3.75530481705680362889110902094, 4.66672123451459212142344880481, 5.42832276623074742721237420550, 6.11826238957396227785655637585, 6.89994642339545498102283322245, 7.55609912020355969196380780205, 8.145678646362832718573957535238