| L(s) = 1 | − 0.559·3-s + 5-s − 0.837·7-s − 2.68·9-s + 0.639·11-s + 4.84·13-s − 0.559·15-s + 6.68·17-s + 0.235·19-s + 0.468·21-s − 0.818·23-s + 25-s + 3.18·27-s + 2.81·29-s − 6.87·31-s − 0.357·33-s − 0.837·35-s + 2.74·37-s − 2.71·39-s + 2.38·41-s + 7.69·43-s − 2.68·45-s − 5.45·47-s − 6.29·49-s − 3.74·51-s − 10.3·53-s + 0.639·55-s + ⋯ |
| L(s) = 1 | − 0.323·3-s + 0.447·5-s − 0.316·7-s − 0.895·9-s + 0.192·11-s + 1.34·13-s − 0.144·15-s + 1.62·17-s + 0.0541·19-s + 0.102·21-s − 0.170·23-s + 0.200·25-s + 0.612·27-s + 0.523·29-s − 1.23·31-s − 0.0622·33-s − 0.141·35-s + 0.451·37-s − 0.434·39-s + 0.372·41-s + 1.17·43-s − 0.400·45-s − 0.795·47-s − 0.899·49-s − 0.524·51-s − 1.41·53-s + 0.0861·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898354401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898354401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 0.559T + 3T^{2} \) |
| 7 | \( 1 + 0.837T + 7T^{2} \) |
| 11 | \( 1 - 0.639T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 0.235T + 19T^{2} \) |
| 23 | \( 1 + 0.818T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 - 7.69T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 - 0.0293T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 0.501T + 97T^{2} \) |
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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
−8.186633871356520758610864774202, −7.34294246885319285683631912149, −6.47687438691315981565396939697, −5.77857250783579720409843051505, −5.58907404857090438428361585372, −4.46284949044088112247506990517, −3.46638387902688804274440984989, −2.97333166085683601549780080912, −1.72075217165092420630561445022, −0.76120050758947251854701643927,
0.76120050758947251854701643927, 1.72075217165092420630561445022, 2.97333166085683601549780080912, 3.46638387902688804274440984989, 4.46284949044088112247506990517, 5.58907404857090438428361585372, 5.77857250783579720409843051505, 6.47687438691315981565396939697, 7.34294246885319285683631912149, 8.186633871356520758610864774202
