| L(s) = 1 | + 2.60·3-s + 0.183·5-s + 3.24·7-s + 3.77·9-s + 2.13·11-s + 1.69·13-s + 0.478·15-s − 17-s + 3.65·19-s + 8.44·21-s + 6.53·23-s − 4.96·25-s + 2.01·27-s + 3.31·29-s − 4.57·31-s + 5.56·33-s + 0.596·35-s − 3.89·37-s + 4.41·39-s + 7.20·41-s − 2.66·43-s + 0.694·45-s − 8.38·47-s + 3.53·49-s − 2.60·51-s − 8.06·53-s + 0.393·55-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.0822·5-s + 1.22·7-s + 1.25·9-s + 0.644·11-s + 0.470·13-s + 0.123·15-s − 0.242·17-s + 0.837·19-s + 1.84·21-s + 1.36·23-s − 0.993·25-s + 0.388·27-s + 0.615·29-s − 0.822·31-s + 0.968·33-s + 0.100·35-s − 0.639·37-s + 0.706·39-s + 1.12·41-s − 0.405·43-s + 0.103·45-s − 1.22·47-s + 0.504·49-s − 0.364·51-s − 1.10·53-s + 0.0529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.354788493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.354788493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 0.183T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 + 8.38T + 47T^{2} \) |
| 53 | \( 1 + 8.06T + 53T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 7.08T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.515984042887011093892306273580, −7.79387024960442442195241251370, −7.32415682223545461073712702270, −6.36912012297935682045844866101, −5.30218688138580048282687728050, −4.54261055982787147350125317266, −3.68614223549001819854250837540, −3.00136450854780866580422190442, −1.95083684483585334010116150391, −1.28993669033450178955637233235,
1.28993669033450178955637233235, 1.95083684483585334010116150391, 3.00136450854780866580422190442, 3.68614223549001819854250837540, 4.54261055982787147350125317266, 5.30218688138580048282687728050, 6.36912012297935682045844866101, 7.32415682223545461073712702270, 7.79387024960442442195241251370, 8.515984042887011093892306273580
