L(s) = 1 | + 1.84·3-s + 2.40·5-s − 7-s + 0.388·9-s − 5.87·11-s − 6.24·13-s + 4.43·15-s − 5.42·17-s + 2.23·19-s − 1.84·21-s + 23-s + 0.804·25-s − 4.80·27-s + 0.642·29-s − 7.84·31-s − 10.8·33-s − 2.40·35-s + 0.557·37-s − 11.5·39-s + 2.56·41-s + 8.81·43-s + 0.935·45-s − 4.26·47-s + 49-s − 9.98·51-s + 3.01·53-s − 14.1·55-s + ⋯ |
L(s) = 1 | + 1.06·3-s + 1.07·5-s − 0.377·7-s + 0.129·9-s − 1.77·11-s − 1.73·13-s + 1.14·15-s − 1.31·17-s + 0.513·19-s − 0.401·21-s + 0.208·23-s + 0.160·25-s − 0.925·27-s + 0.119·29-s − 1.40·31-s − 1.88·33-s − 0.407·35-s + 0.0916·37-s − 1.84·39-s + 0.401·41-s + 1.34·43-s + 0.139·45-s − 0.622·47-s + 0.142·49-s − 1.39·51-s + 0.414·53-s − 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 29 | \( 1 - 0.642T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 - 0.557T + 37T^{2} \) |
| 41 | \( 1 - 2.56T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 + 0.148T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 0.861T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.646625644125071607952671127602, −7.60619650280714676134923600979, −7.32999262553913725996105779246, −6.10263353782957353961127829658, −5.35217938106872138756259176811, −4.64969656781408981710177619461, −3.29692880438543948317520096813, −2.37240294059899766754901060629, −2.23057718972739939331513908560, 0,
2.23057718972739939331513908560, 2.37240294059899766754901060629, 3.29692880438543948317520096813, 4.64969656781408981710177619461, 5.35217938106872138756259176811, 6.10263353782957353961127829658, 7.32999262553913725996105779246, 7.60619650280714676134923600979, 8.646625644125071607952671127602