| L(s) = 1 | − 3·5-s + 4·11-s − 4·13-s − 6·17-s + 7·19-s − 2·23-s + 4·25-s − 6·29-s − 2·31-s − 37-s + 11·41-s − 8·43-s − 4·47-s + 3·53-s − 12·55-s + 12·59-s + 2·61-s + 12·65-s + 3·67-s − 11·71-s − 4·73-s − 12·79-s + 9·83-s + 18·85-s − 4·89-s − 21·95-s + 3·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.20·11-s − 1.10·13-s − 1.45·17-s + 1.60·19-s − 0.417·23-s + 4/5·25-s − 1.11·29-s − 0.359·31-s − 0.164·37-s + 1.71·41-s − 1.21·43-s − 0.583·47-s + 0.412·53-s − 1.61·55-s + 1.56·59-s + 0.256·61-s + 1.48·65-s + 0.366·67-s − 1.30·71-s − 0.468·73-s − 1.35·79-s + 0.987·83-s + 1.95·85-s − 0.423·89-s − 2.15·95-s + 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.75135784877808, −13.09518030527665, −12.81086513391278, −11.97307333918390, −11.87405054773569, −11.34174715883585, −11.18005020639437, −10.29866814525267, −9.774582901327339, −9.327394779815737, −8.846421387559491, −8.378868107731046, −7.661017739707491, −7.314293516657816, −7.026080798535479, −6.357727281008637, −5.707801832957048, −5.053055476358912, −4.537776971498463, −3.983633776260169, −3.652494155256093, −2.964469935593893, −2.251175827125831, −1.557370409341549, −0.6843540479870546, 0,
0.6843540479870546, 1.557370409341549, 2.251175827125831, 2.964469935593893, 3.652494155256093, 3.983633776260169, 4.537776971498463, 5.053055476358912, 5.707801832957048, 6.357727281008637, 7.026080798535479, 7.314293516657816, 7.661017739707491, 8.378868107731046, 8.846421387559491, 9.327394779815737, 9.774582901327339, 10.29866814525267, 11.18005020639437, 11.34174715883585, 11.87405054773569, 11.97307333918390, 12.81086513391278, 13.09518030527665, 13.75135784877808
