| L(s) = 1 | − 0.210·2-s − 0.138·3-s − 1.95·4-s + 1.89·5-s + 0.0291·6-s + 4.57·7-s + 0.831·8-s − 2.98·9-s − 0.398·10-s − 3.88·11-s + 0.271·12-s − 0.961·14-s − 0.262·15-s + 3.73·16-s − 2.27·17-s + 0.626·18-s − 5.91·19-s − 3.70·20-s − 0.634·21-s + 0.817·22-s + 0.275·23-s − 0.115·24-s − 1.41·25-s + 0.829·27-s − 8.94·28-s + 5.82·29-s + 0.0552·30-s + ⋯ |
| L(s) = 1 | − 0.148·2-s − 0.0800·3-s − 0.977·4-s + 0.847·5-s + 0.0119·6-s + 1.72·7-s + 0.294·8-s − 0.993·9-s − 0.126·10-s − 1.17·11-s + 0.0782·12-s − 0.257·14-s − 0.0678·15-s + 0.934·16-s − 0.551·17-s + 0.147·18-s − 1.35·19-s − 0.828·20-s − 0.138·21-s + 0.174·22-s + 0.0573·23-s − 0.0235·24-s − 0.282·25-s + 0.159·27-s − 1.69·28-s + 1.08·29-s + 0.0100·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473644194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473644194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.210T + 2T^{2} \) |
| 3 | \( 1 + 0.138T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 - 0.275T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 9.20T + 71T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.372616516431600027951432606448, −7.86776251853824902241899196614, −6.72378781012869079588464216761, −5.75236433019441465029057343967, −5.23880754797061177972326629558, −4.75248244252021471582250125947, −3.92139402500521076580012395352, −2.53281868435529891289920427398, −1.98078651104261044620024602165, −0.67211588943971783694933409857,
0.67211588943971783694933409857, 1.98078651104261044620024602165, 2.53281868435529891289920427398, 3.92139402500521076580012395352, 4.75248244252021471582250125947, 5.23880754797061177972326629558, 5.75236433019441465029057343967, 6.72378781012869079588464216761, 7.86776251853824902241899196614, 8.372616516431600027951432606448
