L(s) = 1 | − 3-s + 3·5-s − 2·9-s + 4·11-s − 5·13-s − 3·15-s − 5·17-s + 23-s + 4·25-s + 5·27-s + 3·29-s − 4·31-s − 4·33-s + 2·37-s + 5·39-s − 5·41-s + 11·43-s − 6·45-s + 5·47-s − 7·49-s + 5·51-s − 9·53-s + 12·55-s − 13·59-s − 61-s − 15·65-s + 5·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s + 1.20·11-s − 1.38·13-s − 0.774·15-s − 1.21·17-s + 0.208·23-s + 4/5·25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.800·39-s − 0.780·41-s + 1.67·43-s − 0.894·45-s + 0.729·47-s − 49-s + 0.700·51-s − 1.23·53-s + 1.61·55-s − 1.69·59-s − 0.128·61-s − 1.86·65-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.59872473365232218068791948934, −6.83075044054572343025756669507, −6.23902962294964096714433558555, −5.75931851189139969458779396269, −4.91215892937544006745154073668, −4.34718693991791536259345999078, −3.02054133401327408268063812218, −2.28887187412560605052544112344, −1.40646902110043167226315148861, 0,
1.40646902110043167226315148861, 2.28887187412560605052544112344, 3.02054133401327408268063812218, 4.34718693991791536259345999078, 4.91215892937544006745154073668, 5.75931851189139969458779396269, 6.23902962294964096714433558555, 6.83075044054572343025756669507, 7.59872473365232218068791948934