| L(s) = 1 | + 3-s + 5-s + 9-s − 11-s + 15-s − 8·17-s + 4·19-s − 4·23-s − 4·25-s + 27-s − 5·29-s − 7·31-s − 33-s + 8·37-s + 4·41-s − 10·43-s + 45-s − 6·47-s − 8·51-s − 53-s − 55-s + 4·57-s − 9·59-s − 2·61-s − 2·67-s − 4·69-s − 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.258·15-s − 1.94·17-s + 0.917·19-s − 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.928·29-s − 1.25·31-s − 0.174·33-s + 1.31·37-s + 0.624·41-s − 1.52·43-s + 0.149·45-s − 0.875·47-s − 1.12·51-s − 0.137·53-s − 0.134·55-s + 0.529·57-s − 1.17·59-s − 0.256·61-s − 0.244·67-s − 0.481·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.900145213689744905789782519690, −7.35768119493839498271855908380, −6.46868875646122403712660326716, −5.82445986287948043471307085364, −4.91504819070255871343944639312, −4.14000594489856451107349881246, −3.29947445619427324877889267262, −2.29935079307391249319915379929, −1.69312339558267674983238812124, 0,
1.69312339558267674983238812124, 2.29935079307391249319915379929, 3.29947445619427324877889267262, 4.14000594489856451107349881246, 4.91504819070255871343944639312, 5.82445986287948043471307085364, 6.46868875646122403712660326716, 7.35768119493839498271855908380, 7.900145213689744905789782519690
