| L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 13-s − 2·17-s − 21-s + 8·23-s − 5·25-s − 27-s + 2·29-s + 4·31-s − 2·33-s + 6·37-s + 39-s − 8·41-s + 4·43-s − 6·47-s + 49-s + 2·51-s − 2·53-s + 2·59-s + 2·61-s + 63-s + 4·67-s − 8·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.218·21-s + 1.66·23-s − 25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.986·37-s + 0.160·39-s − 1.24·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s + 0.260·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s − 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638854305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638854305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.343703928733587903115238745948, −7.59360391319494946702443610315, −6.77952917202325686841418936376, −6.29734392191485410135265413331, −5.31563948412334827219518652163, −4.72806218691344613451679988179, −3.95149052690476985898155872116, −2.90633883207775464846897147288, −1.81705603263315543974257701492, −0.76208878248137137011420618317,
0.76208878248137137011420618317, 1.81705603263315543974257701492, 2.90633883207775464846897147288, 3.95149052690476985898155872116, 4.72806218691344613451679988179, 5.31563948412334827219518652163, 6.29734392191485410135265413331, 6.77952917202325686841418936376, 7.59360391319494946702443610315, 8.343703928733587903115238745948
