| L(s) = 1 | − 0.271·2-s + 3-s − 1.92·4-s − 0.271·6-s + 0.253·7-s + 1.06·8-s + 9-s − 4.86·11-s − 1.92·12-s − 5.29·13-s − 0.0687·14-s + 3.56·16-s − 1.56·17-s − 0.271·18-s − 4.91·19-s + 0.253·21-s + 1.31·22-s − 0.469·23-s + 1.06·24-s + 1.43·26-s + 27-s − 0.488·28-s + 3.24·29-s + 5.91·31-s − 3.09·32-s − 4.86·33-s + 0.423·34-s + ⋯ |
| L(s) = 1 | − 0.191·2-s + 0.577·3-s − 0.963·4-s − 0.110·6-s + 0.0957·7-s + 0.376·8-s + 0.333·9-s − 1.46·11-s − 0.556·12-s − 1.46·13-s − 0.0183·14-s + 0.891·16-s − 0.378·17-s − 0.0639·18-s − 1.12·19-s + 0.0553·21-s + 0.281·22-s − 0.0979·23-s + 0.217·24-s + 0.281·26-s + 0.192·27-s − 0.0922·28-s + 0.601·29-s + 1.06·31-s − 0.547·32-s − 0.846·33-s + 0.0726·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.075661904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075661904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.271T + 2T^{2} \) |
| 7 | \( 1 - 0.253T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 0.469T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 5.74T + 43T^{2} \) |
| 53 | \( 1 + 0.162T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 7.42T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 7.19T + 89T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
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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
−8.397397391849287133283555019913, −8.025698124057229079284893484562, −7.40839816873159261909683178962, −6.38108803686962628963768389816, −5.31703344420224644950295952278, −4.68692336626527687989841854914, −4.10257247855983769711405276648, −2.80282579819016271679306145590, −2.25073623488423650976784744826, −0.59022397572498046050868303976,
0.59022397572498046050868303976, 2.25073623488423650976784744826, 2.80282579819016271679306145590, 4.10257247855983769711405276648, 4.68692336626527687989841854914, 5.31703344420224644950295952278, 6.38108803686962628963768389816, 7.40839816873159261909683178962, 8.025698124057229079284893484562, 8.397397391849287133283555019913
