L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 2·13-s − 14-s + 16-s − 2·17-s − 19-s − 2·20-s − 8·23-s − 25-s + 2·26-s − 28-s − 2·29-s + 4·31-s + 32-s − 2·34-s + 2·35-s + 2·37-s − 38-s − 2·40-s − 6·41-s − 12·43-s − 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.328·37-s − 0.162·38-s − 0.316·40-s − 0.937·41-s − 1.82·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.281501563887625996464501642974, −7.912508640806354969373892342740, −6.85350902861062329638401661291, −6.28352954329121793129107491158, −5.40960412801561297413267643753, −4.33084839581572636702073752349, −3.84907283320357724315521311959, −2.94285784851423154213390802202, −1.74411672296421629506581859944, 0,
1.74411672296421629506581859944, 2.94285784851423154213390802202, 3.84907283320357724315521311959, 4.33084839581572636702073752349, 5.40960412801561297413267643753, 6.28352954329121793129107491158, 6.85350902861062329638401661291, 7.912508640806354969373892342740, 8.281501563887625996464501642974