| L(s) = 1 | − 2.30·2-s + 3.30·4-s + 7-s − 3.00·8-s + 3·11-s − 2.60·13-s − 2.30·14-s + 0.302·16-s − 4.60·17-s + 6.60·19-s − 6.90·22-s + 6.21·23-s + 6·26-s + 3.30·28-s + 7.60·29-s − 7.21·31-s + 5.30·32-s + 10.6·34-s − 4.21·37-s − 15.2·38-s + 9.60·43-s + 9.90·44-s − 14.3·46-s − 10.6·47-s + 49-s − 8.60·52-s − 3.21·53-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s + 0.377·7-s − 1.06·8-s + 0.904·11-s − 0.722·13-s − 0.615·14-s + 0.0756·16-s − 1.11·17-s + 1.51·19-s − 1.47·22-s + 1.29·23-s + 1.17·26-s + 0.624·28-s + 1.41·29-s − 1.29·31-s + 0.937·32-s + 1.81·34-s − 0.692·37-s − 2.46·38-s + 1.46·43-s + 1.49·44-s − 2.10·46-s − 1.54·47-s + 0.142·49-s − 1.19·52-s − 0.441·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8076821183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8076821183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 0.605T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 0.788T + 97T^{2} \) |
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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
−9.373305601241735506001126093248, −8.764442251009763260469150801085, −8.026108328558568297157779727521, −7.02542870717423248568970136929, −6.83007513077537615270558567408, −5.40939092430265726654078949254, −4.43774547754851425401365398106, −3.03268511837486717539072167919, −1.89105327039571605253244359212, −0.831712353303606640939264466187,
0.831712353303606640939264466187, 1.89105327039571605253244359212, 3.03268511837486717539072167919, 4.43774547754851425401365398106, 5.40939092430265726654078949254, 6.83007513077537615270558567408, 7.02542870717423248568970136929, 8.026108328558568297157779727521, 8.764442251009763260469150801085, 9.373305601241735506001126093248
