| L(s) = 1 | − 2.23·2-s + 3-s + 2.97·4-s + 2.00·5-s − 2.23·6-s − 4.12·7-s − 2.17·8-s + 9-s − 4.48·10-s + 3.38·11-s + 2.97·12-s + 13-s + 9.19·14-s + 2.00·15-s − 1.09·16-s − 1.97·17-s − 2.23·18-s − 5.67·19-s + 5.98·20-s − 4.12·21-s − 7.54·22-s + 5.94·23-s − 2.17·24-s − 0.960·25-s − 2.23·26-s + 27-s − 12.2·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.48·4-s + 0.898·5-s − 0.910·6-s − 1.55·7-s − 0.769·8-s + 0.333·9-s − 1.41·10-s + 1.01·11-s + 0.858·12-s + 0.277·13-s + 2.45·14-s + 0.518·15-s − 0.274·16-s − 0.479·17-s − 0.525·18-s − 1.30·19-s + 1.33·20-s − 0.899·21-s − 1.60·22-s + 1.23·23-s − 0.444·24-s − 0.192·25-s − 0.437·26-s + 0.192·27-s − 2.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035747055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035747055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 - 4.45T + 31T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 5.77T + 71T^{2} \) |
| 73 | \( 1 - 5.76T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 + 1.02T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−8.725304252272403171347287152460, −7.986647882780958564673107080558, −6.87828719839874133006653938430, −6.59466034503607633728018892718, −6.01250010558868749643570367131, −4.54993928046273942020102201298, −3.53790269922407871443714029329, −2.59469296099172716821653913785, −1.83359306519185747804823537503, −0.71767789077212085914981889863,
0.71767789077212085914981889863, 1.83359306519185747804823537503, 2.59469296099172716821653913785, 3.53790269922407871443714029329, 4.54993928046273942020102201298, 6.01250010558868749643570367131, 6.59466034503607633728018892718, 6.87828719839874133006653938430, 7.986647882780958564673107080558, 8.725304252272403171347287152460
