| L(s) = 1 | − 3-s + 9-s − 5·11-s − 2·13-s − 4·17-s + 6·19-s + 23-s − 27-s + 29-s + 2·31-s + 5·33-s + 5·37-s + 2·39-s + 7·43-s − 2·47-s + 4·51-s − 6·53-s − 6·57-s − 6·59-s − 3·67-s − 69-s − 9·71-s + 12·73-s − 3·79-s + 81-s + 6·83-s − 87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s + 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.359·31-s + 0.870·33-s + 0.821·37-s + 0.320·39-s + 1.06·43-s − 0.291·47-s + 0.560·51-s − 0.824·53-s − 0.794·57-s − 0.781·59-s − 0.366·67-s − 0.120·69-s − 1.06·71-s + 1.40·73-s − 0.337·79-s + 1/9·81-s + 0.658·83-s − 0.107·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.58735403829794, −14.02097603908343, −13.44305310923759, −13.10766587642502, −12.55693167003464, −12.07338705921824, −11.48699150908229, −11.03936897579808, −10.53812175572242, −10.09311985690521, −9.447585133918911, −9.099227362644718, −8.208310013141925, −7.729939966870240, −7.384585335744920, −6.682473705844853, −6.108185897615886, −5.492916700597101, −4.978684568975749, −4.608150108180575, −3.829776851115951, −2.873464837110734, −2.617250102460411, −1.691804586675551, −0.7792044067789580, 0,
0.7792044067789580, 1.691804586675551, 2.617250102460411, 2.873464837110734, 3.829776851115951, 4.608150108180575, 4.978684568975749, 5.492916700597101, 6.108185897615886, 6.682473705844853, 7.384585335744920, 7.729939966870240, 8.208310013141925, 9.099227362644718, 9.447585133918911, 10.09311985690521, 10.53812175572242, 11.03936897579808, 11.48699150908229, 12.07338705921824, 12.55693167003464, 13.10766587642502, 13.44305310923759, 14.02097603908343, 14.58735403829794
