| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·13-s + 2·15-s − 2·17-s − 4·19-s − 25-s + 27-s − 6·29-s − 6·37-s + 2·39-s + 6·41-s + 8·43-s + 2·45-s + 8·47-s − 2·51-s − 6·53-s − 4·57-s + 12·59-s + 10·61-s + 4·65-s + 16·67-s + 8·71-s + 6·73-s − 75-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s + 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.95·67-s + 0.949·71-s + 0.702·73-s − 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.149383564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.149383564\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.72603143607158752044686741875, −7.01457138950833382896839978793, −6.30733389772304530213003727973, −5.74071340127034582561549161433, −4.99008942844597342041954417920, −4.02541501977557652775434352579, −3.54527166050418800052699168253, −2.20363931114562236079736795767, −2.15422604112595950745863621955, −0.820749216643845756760176902569,
0.820749216643845756760176902569, 2.15422604112595950745863621955, 2.20363931114562236079736795767, 3.54527166050418800052699168253, 4.02541501977557652775434352579, 4.99008942844597342041954417920, 5.74071340127034582561549161433, 6.30733389772304530213003727973, 7.01457138950833382896839978793, 7.72603143607158752044686741875
