L(s) = 1 | − 0.801·2-s + 3.24·3-s − 1.35·4-s + 2·5-s − 2.60·6-s + 1.19·7-s + 2.69·8-s + 7.54·9-s − 1.60·10-s − 0.335·11-s − 4.40·12-s + 5.04·13-s − 0.960·14-s + 6.49·15-s + 0.554·16-s − 1.60·17-s − 6.04·18-s − 0.951·19-s − 2.71·20-s + 3.89·21-s + 0.268·22-s + 3.55·23-s + 8.74·24-s − 25-s − 4.04·26-s + 14.7·27-s − 1.62·28-s + ⋯ |
L(s) = 1 | − 0.567·2-s + 1.87·3-s − 0.678·4-s + 0.894·5-s − 1.06·6-s + 0.452·7-s + 0.951·8-s + 2.51·9-s − 0.507·10-s − 0.101·11-s − 1.27·12-s + 1.40·13-s − 0.256·14-s + 1.67·15-s + 0.138·16-s − 0.388·17-s − 1.42·18-s − 0.218·19-s − 0.606·20-s + 0.848·21-s + 0.0572·22-s + 0.741·23-s + 1.78·24-s − 0.200·25-s − 0.794·26-s + 2.83·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.903414665\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.903414665\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 0.951T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 47 | \( 1 + T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 6.65T + 67T^{2} \) |
| 71 | \( 1 - 0.670T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 4.38T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.081688945040256857795794157386, −8.714345928096862435792133394019, −7.933173987616933172605181218081, −7.33839509413845490638973917188, −6.16460409290444967940956953888, −4.99020327105149995936924047487, −4.04949615803069201683721008308, −3.30262268133220180229966303419, −2.02937746943709034555588717820, −1.39473037679144189573189720509,
1.39473037679144189573189720509, 2.02937746943709034555588717820, 3.30262268133220180229966303419, 4.04949615803069201683721008308, 4.99020327105149995936924047487, 6.16460409290444967940956953888, 7.33839509413845490638973917188, 7.933173987616933172605181218081, 8.714345928096862435792133394019, 9.081688945040256857795794157386