| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·11-s − 5·13-s + 14-s + 16-s + 6·17-s + 6·19-s − 2·22-s − 8·23-s + 5·26-s − 28-s − 5·29-s + 7·31-s − 32-s − 6·34-s − 4·37-s − 6·38-s + 41-s − 4·43-s + 2·44-s + 8·46-s − 2·47-s + 49-s − 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s − 0.426·22-s − 1.66·23-s + 0.980·26-s − 0.188·28-s − 0.928·29-s + 1.25·31-s − 0.176·32-s − 1.02·34-s − 0.657·37-s − 0.973·38-s + 0.156·41-s − 0.609·43-s + 0.301·44-s + 1.17·46-s − 0.291·47-s + 1/7·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221683022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221683022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.80086822962811718158955057848, −7.14113282289659836705180531226, −6.50897099899862797242229256002, −5.64481603058124069509657863206, −5.16736915433467683967326320419, −4.04172550749519698595893040126, −3.34125832838287653114190940385, −2.53270703302635235619341365443, −1.60663000541929431436322235197, −0.59816612814495419367909923286,
0.59816612814495419367909923286, 1.60663000541929431436322235197, 2.53270703302635235619341365443, 3.34125832838287653114190940385, 4.04172550749519698595893040126, 5.16736915433467683967326320419, 5.64481603058124069509657863206, 6.50897099899862797242229256002, 7.14113282289659836705180531226, 7.80086822962811718158955057848
