| L(s) = 1 | + 5-s − 2·11-s + 5·17-s − 2·19-s − 2·23-s − 4·25-s − 10·29-s + 5·37-s + 3·41-s − 7·43-s − 3·47-s − 6·53-s − 2·55-s + 59-s + 6·61-s + 4·67-s − 8·71-s − 10·73-s − 3·79-s − 13·83-s + 5·85-s − 6·89-s − 2·95-s + 14·97-s + 10·101-s − 2·103-s + 18·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s + 1.21·17-s − 0.458·19-s − 0.417·23-s − 4/5·25-s − 1.85·29-s + 0.821·37-s + 0.468·41-s − 1.06·43-s − 0.437·47-s − 0.824·53-s − 0.269·55-s + 0.130·59-s + 0.768·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.337·79-s − 1.42·83-s + 0.542·85-s − 0.635·89-s − 0.205·95-s + 1.42·97-s + 0.995·101-s − 0.197·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.76456981734616093597728455123, −7.29746063159730572693838026716, −6.19868709410160924353311472175, −5.73696393636086294074076174532, −5.03657247790225663328782091602, −4.06228954105137078888083784226, −3.28369907289621354077175464702, −2.31209905815630362408369439963, −1.46599510921028086426564310651, 0,
1.46599510921028086426564310651, 2.31209905815630362408369439963, 3.28369907289621354077175464702, 4.06228954105137078888083784226, 5.03657247790225663328782091602, 5.73696393636086294074076174532, 6.19868709410160924353311472175, 7.29746063159730572693838026716, 7.76456981734616093597728455123
