L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·13-s − 14-s + 16-s − 6·17-s + 19-s − 6·23-s − 5·25-s − 2·26-s + 28-s + 6·29-s − 10·31-s − 32-s + 6·34-s + 8·37-s − 38-s − 6·41-s − 4·43-s + 6·46-s + 49-s + 5·50-s + 2·52-s − 6·53-s − 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 1.25·23-s − 25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s + 1.31·37-s − 0.162·38-s − 0.937·41-s − 0.609·43-s + 0.884·46-s + 1/7·49-s + 0.707·50-s + 0.277·52-s − 0.824·53-s − 0.133·56-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.489842326698717837398920644057, −8.034923960614401653103382775527, −7.12067428738848652227633051995, −6.36808655516974190816452684696, −5.61517922411602047981813256221, −4.50864903583377422386297334125, −3.66111055125572467928556646511, −2.40183884965549070769066097540, −1.55216464217009888725261702980, 0,
1.55216464217009888725261702980, 2.40183884965549070769066097540, 3.66111055125572467928556646511, 4.50864903583377422386297334125, 5.61517922411602047981813256221, 6.36808655516974190816452684696, 7.12067428738848652227633051995, 8.034923960614401653103382775527, 8.489842326698717837398920644057