L(s) = 1 | + 5-s − 4.53·7-s − 5.89·11-s − 13-s + 2.53·17-s + 1.35·19-s + 5.25·23-s + 25-s + 0.643·29-s − 2·31-s − 4.53·35-s − 3.89·37-s − 3.89·41-s − 3.79·43-s − 3.79·47-s + 13.6·49-s + 2.10·53-s − 5.89·55-s − 4·59-s + 0.103·61-s − 65-s − 5.79·67-s − 3.18·71-s + 10.4·73-s + 26.7·77-s + 10.9·79-s − 4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.71·7-s − 1.77·11-s − 0.277·13-s + 0.615·17-s + 0.311·19-s + 1.09·23-s + 0.200·25-s + 0.119·29-s − 0.359·31-s − 0.767·35-s − 0.640·37-s − 0.608·41-s − 0.578·43-s − 0.553·47-s + 1.94·49-s + 0.288·53-s − 0.795·55-s − 0.520·59-s + 0.0132·61-s − 0.124·65-s − 0.707·67-s − 0.377·71-s + 1.22·73-s + 3.05·77-s + 1.23·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082918014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082918014\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 0.643T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 0.103T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 3.18T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−8.296833347139894506307983639669, −7.46123920667193056775062248043, −6.88779816628616842068377915231, −6.09779973392124751152867789980, −5.39977481540945092528088553814, −4.80903981287114003742674158190, −3.35302585716802395895543196380, −3.08040714273609987619151849306, −2.10571219292078273593610065945, −0.54419537618240490971657655598,
0.54419537618240490971657655598, 2.10571219292078273593610065945, 3.08040714273609987619151849306, 3.35302585716802395895543196380, 4.80903981287114003742674158190, 5.39977481540945092528088553814, 6.09779973392124751152867789980, 6.88779816628616842068377915231, 7.46123920667193056775062248043, 8.296833347139894506307983639669