| L(s) = 1 | + 2.66·2-s − 1.82·3-s + 5.09·4-s − 1.36·5-s − 4.86·6-s + 0.192·7-s + 8.23·8-s + 0.334·9-s − 3.64·10-s + 11-s − 9.29·12-s + 1.93·13-s + 0.512·14-s + 2.49·15-s + 11.7·16-s + 2.15·17-s + 0.891·18-s − 6.96·20-s − 0.351·21-s + 2.66·22-s + 2.66·23-s − 15.0·24-s − 3.12·25-s + 5.16·26-s + 4.86·27-s + 0.979·28-s − 10.1·29-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 1.05·3-s + 2.54·4-s − 0.611·5-s − 1.98·6-s + 0.0727·7-s + 2.91·8-s + 0.111·9-s − 1.15·10-s + 0.301·11-s − 2.68·12-s + 0.538·13-s + 0.136·14-s + 0.644·15-s + 2.93·16-s + 0.521·17-s + 0.210·18-s − 1.55·20-s − 0.0766·21-s + 0.567·22-s + 0.556·23-s − 3.06·24-s − 0.625·25-s + 1.01·26-s + 0.936·27-s + 0.185·28-s − 1.89·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.348481881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.348481881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 0.192T + 7T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 0.568T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 8.27T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 0.963T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.993891456192566923062625032628, −7.44936185341041237807252170889, −6.55463034759847492080231487169, −5.98492372971616364472475344210, −5.50927878622559866672741885199, −4.68148364560363641971132670745, −4.03585780520763248969998019851, −3.33951021657688304989089571369, −2.32676380962450730497981884878, −0.980506032160636117467575452157,
0.980506032160636117467575452157, 2.32676380962450730497981884878, 3.33951021657688304989089571369, 4.03585780520763248969998019851, 4.68148364560363641971132670745, 5.50927878622559866672741885199, 5.98492372971616364472475344210, 6.55463034759847492080231487169, 7.44936185341041237807252170889, 7.993891456192566923062625032628
