| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 2·11-s − 15-s + 5·17-s + 2·19-s + 2·21-s − 6·23-s − 4·25-s + 27-s − 9·29-s + 4·31-s − 2·33-s − 2·35-s − 11·37-s + 5·41-s − 10·43-s − 45-s − 2·47-s − 3·49-s + 5·51-s − 53-s + 2·55-s + 2·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.258·15-s + 1.21·17-s + 0.458·19-s + 0.436·21-s − 1.25·23-s − 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s − 1.80·37-s + 0.780·41-s − 1.52·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.700·51-s − 0.137·53-s + 0.269·55-s + 0.264·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.64423226564212043595271652374, −7.08142295176549532047052350697, −5.96424039769385255063720529534, −5.36479529989294717888715278836, −4.63615231186923164931224148646, −3.73497568586928780605657211420, −3.26392646415919491515479019158, −2.14052498833297376597850353811, −1.45933009628315391298305498468, 0,
1.45933009628315391298305498468, 2.14052498833297376597850353811, 3.26392646415919491515479019158, 3.73497568586928780605657211420, 4.63615231186923164931224148646, 5.36479529989294717888715278836, 5.96424039769385255063720529534, 7.08142295176549532047052350697, 7.64423226564212043595271652374
