L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 3·11-s − 2·13-s + 2·14-s + 16-s + 6·17-s + 19-s + 3·22-s + 3·23-s + 2·26-s − 2·28-s − 3·29-s − 7·31-s − 32-s − 6·34-s − 2·37-s − 38-s + 6·41-s + 10·43-s − 3·44-s − 3·46-s + 12·47-s − 3·49-s − 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.639·22-s + 0.625·23-s + 0.392·26-s − 0.377·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.162·38-s + 0.937·41-s + 1.52·43-s − 0.452·44-s − 0.442·46-s + 1.75·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.56256793617362073104917922739, −7.03066416163906762692161294169, −5.96303074051700828447950084533, −5.61026960006077446093623966160, −4.72812632635255536675417439985, −3.60595096490209094182210284905, −2.99344629078656661974615421767, −2.21934814903521977949044920875, −1.05916255981662646341373655246, 0,
1.05916255981662646341373655246, 2.21934814903521977949044920875, 2.99344629078656661974615421767, 3.60595096490209094182210284905, 4.72812632635255536675417439985, 5.61026960006077446093623966160, 5.96303074051700828447950084533, 7.03066416163906762692161294169, 7.56256793617362073104917922739