L(s) = 1 | − 2-s − 3-s + 4-s − 3.21·5-s + 6-s − 8-s + 9-s + 3.21·10-s − 5.66·11-s − 12-s + 1.33·13-s + 3.21·15-s + 16-s + 1.56·17-s − 18-s + 19-s − 3.21·20-s + 5.66·22-s + 3.45·23-s + 24-s + 5.36·25-s − 1.33·26-s − 27-s + 6.39·29-s − 3.21·30-s − 6.04·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.43·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.01·10-s − 1.70·11-s − 0.288·12-s + 0.369·13-s + 0.831·15-s + 0.250·16-s + 0.380·17-s − 0.235·18-s + 0.229·19-s − 0.719·20-s + 1.20·22-s + 0.720·23-s + 0.204·24-s + 1.07·25-s − 0.261·26-s − 0.192·27-s + 1.18·29-s − 0.587·30-s − 1.08·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3014797673\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3014797673\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 3.21T + 5T^{2} \) |
| 11 | \( 1 + 5.66T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 - 0.463T + 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 + 0.555T + 67T^{2} \) |
| 71 | \( 1 - 0.389T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.141762994852577174750158709858, −7.49018032396512567722530211748, −7.02190619452818753910599389760, −6.09199040841749285174484747776, −5.16307602673016704335792879801, −4.67008140250559109857869047954, −3.47216425122398568668328771235, −2.95805295519383752787733985617, −1.57824287297520251639690212969, −0.33764433985117655155917930613,
0.33764433985117655155917930613, 1.57824287297520251639690212969, 2.95805295519383752787733985617, 3.47216425122398568668328771235, 4.67008140250559109857869047954, 5.16307602673016704335792879801, 6.09199040841749285174484747776, 7.02190619452818753910599389760, 7.49018032396512567722530211748, 8.141762994852577174750158709858