L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s + 6·13-s + 16-s + 6·17-s + 18-s + 4·19-s − 4·22-s − 23-s − 24-s + 6·26-s − 27-s − 6·29-s − 8·31-s + 32-s + 4·33-s + 6·34-s + 36-s − 6·37-s + 4·38-s − 6·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.66·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 1.17·26-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.648·38-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.585328839\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.585328839\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.458061300494682896865914145018, −7.59695486556492623397829921809, −7.16942439345268818873635576021, −5.89144508579493969352637366075, −5.69575389960111068762856091027, −4.99750076301973389853903362391, −3.77390349557881755875178604468, −3.36118387864159083359166925976, −2.05475348294824284760402709946, −0.910928405014543625906670390332,
0.910928405014543625906670390332, 2.05475348294824284760402709946, 3.36118387864159083359166925976, 3.77390349557881755875178604468, 4.99750076301973389853903362391, 5.69575389960111068762856091027, 5.89144508579493969352637366075, 7.16942439345268818873635576021, 7.59695486556492623397829921809, 8.458061300494682896865914145018