| L(s) = 1 | − 2.37·3-s + 4.37·5-s + 2.62·9-s − 10.3·15-s + 1.62·23-s + 14.1·25-s + 0.883·27-s + 11.1·31-s + 5.11·37-s + 11.4·45-s − 12·47-s − 7·49-s + 6·53-s + 10.3·59-s − 15.1·67-s − 3.86·69-s + 15.8·71-s − 33.4·75-s − 9.97·81-s − 9.86·89-s − 26.3·93-s − 17.1·97-s − 4·103-s − 12.1·111-s − 7.62·113-s + 7.11·115-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 1.95·5-s + 0.875·9-s − 2.67·15-s + 0.339·23-s + 2.82·25-s + 0.169·27-s + 1.99·31-s + 0.841·37-s + 1.71·45-s − 1.75·47-s − 49-s + 0.824·53-s + 1.35·59-s − 1.84·67-s − 0.464·69-s + 1.88·71-s − 3.86·75-s − 1.10·81-s − 1.04·89-s − 2.73·93-s − 1.73·97-s − 0.394·103-s − 1.15·111-s − 0.717·113-s + 0.663·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277950890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277950890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 9.86T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−10.90678716645046167437176099668, −10.09890544132751106411658374752, −9.592596935776882217962835034484, −8.404636360958729750555798476685, −6.75894472225758429941547830740, −6.26167928280954426402695091260, −5.44854738660372947623127224204, −4.72417600974870336732018508119, −2.66755610569428205499615310321, −1.23008901880303198978723571138,
1.23008901880303198978723571138, 2.66755610569428205499615310321, 4.72417600974870336732018508119, 5.44854738660372947623127224204, 6.26167928280954426402695091260, 6.75894472225758429941547830740, 8.404636360958729750555798476685, 9.592596935776882217962835034484, 10.09890544132751106411658374752, 10.90678716645046167437176099668
