| L(s) = 1 | − 1.30·5-s + 7-s − 0.605·13-s − 17-s − 0.605·19-s − 3.30·25-s + 0.697·31-s − 1.30·35-s + 4.60·37-s + 6.90·41-s + 3.69·43-s + 2.60·47-s + 49-s + 7.30·53-s + 5.21·59-s + 2.90·61-s + 0.788·65-s − 5.30·67-s − 13.8·71-s + 2.90·73-s + 5.39·79-s − 6·83-s + 1.30·85-s + 9.39·89-s − 0.605·91-s + 0.788·95-s − 2.69·97-s + ⋯ |
| L(s) = 1 | − 0.582·5-s + 0.377·7-s − 0.167·13-s − 0.242·17-s − 0.138·19-s − 0.660·25-s + 0.125·31-s − 0.220·35-s + 0.757·37-s + 1.07·41-s + 0.563·43-s + 0.380·47-s + 0.142·49-s + 1.00·53-s + 0.678·59-s + 0.372·61-s + 0.0978·65-s − 0.647·67-s − 1.63·71-s + 0.340·73-s + 0.606·79-s − 0.658·83-s + 0.141·85-s + 0.995·89-s − 0.0634·91-s + 0.0809·95-s − 0.273·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576257317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576257317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 0.697T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−8.312221170000839073251357970616, −7.65705935279234566340282350770, −7.10389694933936418954119087365, −6.13841348612126649168166510717, −5.46195039768466721318032710249, −4.45682831339284094802878470208, −3.99365513929709601846282306303, −2.90743018769467865146798896447, −1.99354253016083482081161234246, −0.70475026488933325424144263335,
0.70475026488933325424144263335, 1.99354253016083482081161234246, 2.90743018769467865146798896447, 3.99365513929709601846282306303, 4.45682831339284094802878470208, 5.46195039768466721318032710249, 6.13841348612126649168166510717, 7.10389694933936418954119087365, 7.65705935279234566340282350770, 8.312221170000839073251357970616
