| L(s) = 1 | + 3-s + 9-s − 4·13-s − 4·17-s − 4·19-s − 4·23-s − 5·25-s + 27-s + 2·29-s + 8·31-s − 6·37-s − 4·39-s − 12·41-s − 4·43-s − 8·47-s − 4·51-s + 6·53-s − 4·57-s + 12·59-s − 4·61-s + 4·67-s − 4·69-s + 12·71-s − 8·73-s − 5·75-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.970·17-s − 0.917·19-s − 0.834·23-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s − 1.16·47-s − 0.560·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.512·61-s + 0.488·67-s − 0.481·69-s + 1.42·71-s − 0.936·73-s − 0.577·75-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.318001579149289855836057155815, −8.184913935072126797079476432782, −6.92671762276296813999305771272, −6.54773695073555604095145238596, −5.31213745993752385954663014907, −4.51756279179944272117583721138, −3.70735133964840337307130677204, −2.56715844841857154290151914326, −1.85302717988446910203244685234, 0,
1.85302717988446910203244685234, 2.56715844841857154290151914326, 3.70735133964840337307130677204, 4.51756279179944272117583721138, 5.31213745993752385954663014907, 6.54773695073555604095145238596, 6.92671762276296813999305771272, 8.184913935072126797079476432782, 8.318001579149289855836057155815
