| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 2·11-s + 2·13-s + 16-s − 2·17-s + 6·19-s − 20-s + 2·22-s + 4·23-s + 25-s − 2·26-s − 2·31-s − 32-s + 2·34-s + 2·37-s − 6·38-s + 40-s − 10·41-s − 8·43-s − 2·44-s − 4·46-s + 8·47-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.973·38-s + 0.158·40-s − 1.56·41-s − 1.21·43-s − 0.301·44-s − 0.589·46-s + 1.16·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109031088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109031088\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.534223993894932133935300439508, −7.54989395954188160037997215118, −7.20528475175092734810498968650, −6.30835453368910666484606355248, −5.43872560400832123096403739032, −4.71215531335858462495325223484, −3.55974650518840309017563415718, −2.93117589735157766666354392580, −1.77148799114426370056808540256, −0.66281130095645621330697335573,
0.66281130095645621330697335573, 1.77148799114426370056808540256, 2.93117589735157766666354392580, 3.55974650518840309017563415718, 4.71215531335858462495325223484, 5.43872560400832123096403739032, 6.30835453368910666484606355248, 7.20528475175092734810498968650, 7.54989395954188160037997215118, 8.534223993894932133935300439508
