L(s) = 1 | − 5-s − 4·7-s + 2·13-s − 6·17-s + 2·19-s + 23-s + 25-s − 6·29-s − 4·31-s + 4·35-s + 8·37-s − 6·41-s + 8·43-s − 12·47-s + 9·49-s + 6·53-s + 6·59-s − 10·61-s − 2·65-s + 8·67-s + 6·71-s + 2·73-s − 10·79-s + 12·83-s + 6·85-s − 6·89-s − 8·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s + 1.31·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.781·59-s − 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.712·71-s + 0.234·73-s − 1.12·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9747785660\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9747785660\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.511348744562729280220841628695, −7.59364285167981924845250143718, −6.88272774995945306399970033031, −6.32933415016630436674097688106, −5.58414743690763122838651332783, −4.52179776183978086384556961510, −3.71699009524032920517802328202, −3.11009796581965296703483268985, −2.05836789002722838619797179361, −0.53754359514065542070459562379,
0.53754359514065542070459562379, 2.05836789002722838619797179361, 3.11009796581965296703483268985, 3.71699009524032920517802328202, 4.52179776183978086384556961510, 5.58414743690763122838651332783, 6.32933415016630436674097688106, 6.88272774995945306399970033031, 7.59364285167981924845250143718, 8.511348744562729280220841628695