| L(s) = 1 | + 4.28·5-s + 3.81·11-s + 3.28·13-s + 0.810·17-s + 7.09·19-s + 6.47·23-s + 13.3·25-s − 3.81·29-s + 3.28·31-s − 5.76·37-s − 2.09·41-s − 8.76·43-s − 3.33·47-s − 9.86·53-s + 16.3·55-s − 3.47·59-s + 5.95·61-s + 14.0·65-s − 3.52·67-s − 6.05·71-s − 10.3·73-s − 5.14·79-s + 1.71·83-s + 3.47·85-s − 12.5·89-s + 30.4·95-s + 1.04·97-s + ⋯ |
| L(s) = 1 | + 1.91·5-s + 1.14·11-s + 0.911·13-s + 0.196·17-s + 1.62·19-s + 1.35·23-s + 2.67·25-s − 0.707·29-s + 0.590·31-s − 0.947·37-s − 0.327·41-s − 1.33·43-s − 0.486·47-s − 1.35·53-s + 2.20·55-s − 0.452·59-s + 0.762·61-s + 1.74·65-s − 0.430·67-s − 0.718·71-s − 1.21·73-s − 0.578·79-s + 0.187·83-s + 0.377·85-s − 1.32·89-s + 3.12·95-s + 0.106·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.715281172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.715281172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 4.28T + 5T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 0.810T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 - 5.95T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 1.04T + 97T^{2} \) |
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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
−8.462146843149701752286413560043, −7.21501582855071144312321015201, −6.67765370342093292397124067750, −6.01547815691358075321706376052, −5.40559317418453376914614914403, −4.75664869704516272669824300241, −3.47698314619105383044860361710, −2.88542754034505712771169142514, −1.54797725539764706329463087123, −1.29354482214877467853878995036,
1.29354482214877467853878995036, 1.54797725539764706329463087123, 2.88542754034505712771169142514, 3.47698314619105383044860361710, 4.75664869704516272669824300241, 5.40559317418453376914614914403, 6.01547815691358075321706376052, 6.67765370342093292397124067750, 7.21501582855071144312321015201, 8.462146843149701752286413560043
