| L(s) = 1 | − 3-s + 9-s − 5·11-s + 2·13-s − 2·17-s + 6·19-s − 23-s − 27-s − 3·29-s − 4·31-s + 5·33-s − 5·37-s − 2·39-s − 4·41-s − 7·43-s − 10·47-s + 2·51-s + 2·53-s − 6·57-s + 10·59-s + 8·61-s + 7·67-s + 69-s − 3·71-s + 2·73-s − 11·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.208·23-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.870·33-s − 0.821·37-s − 0.320·39-s − 0.624·41-s − 1.06·43-s − 1.45·47-s + 0.280·51-s + 0.274·53-s − 0.794·57-s + 1.30·59-s + 1.02·61-s + 0.855·67-s + 0.120·69-s − 0.356·71-s + 0.234·73-s − 1.23·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.05420708096866, −12.82970842385163, −12.20458988234109, −11.56717029923621, −11.33575041420932, −10.96121922038579, −10.26976622061513, −9.945219740337482, −9.670748983368295, −8.777082582128609, −8.465834139792659, −7.984301268538136, −7.336955968713031, −7.038871198865649, −6.528335540648669, −5.690180527944584, −5.543543224272342, −5.032388152580366, −4.570223044062125, −3.737468771692829, −3.365844906400656, −2.752403808424511, −1.995670870963202, −1.529033253592777, −0.6340867099835869, 0,
0.6340867099835869, 1.529033253592777, 1.995670870963202, 2.752403808424511, 3.365844906400656, 3.737468771692829, 4.570223044062125, 5.032388152580366, 5.543543224272342, 5.690180527944584, 6.528335540648669, 7.038871198865649, 7.336955968713031, 7.984301268538136, 8.465834139792659, 8.777082582128609, 9.670748983368295, 9.945219740337482, 10.26976622061513, 10.96121922038579, 11.33575041420932, 11.56717029923621, 12.20458988234109, 12.82970842385163, 13.05420708096866
