L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 3·11-s − 12-s − 13-s + 15-s + 16-s − 7·17-s − 18-s − 6·19-s − 20-s − 3·22-s + 9·23-s + 24-s + 25-s + 26-s − 27-s − 6·29-s − 30-s − 32-s − 3·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.639·22-s + 1.87·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.176·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.67238897954841, −12.23848388696782, −11.53132242082182, −11.32205522591335, −11.01516890071478, −10.49217390902053, −10.14651399960436, −9.371600988533814, −9.077630899636648, −8.637918874588677, −8.407234347321530, −7.576326876472358, −7.016043129905918, −6.856353393847301, −6.490984509108720, −5.846808352154559, −5.243280032428022, −4.696296329976911, −4.313153108881793, −3.674911947410386, −3.208969901039180, −2.385376211221427, −1.890140941259862, −1.380162073387961, −0.5302620707674791, 0,
0.5302620707674791, 1.380162073387961, 1.890140941259862, 2.385376211221427, 3.208969901039180, 3.674911947410386, 4.313153108881793, 4.696296329976911, 5.243280032428022, 5.846808352154559, 6.490984509108720, 6.856353393847301, 7.016043129905918, 7.576326876472358, 8.407234347321530, 8.637918874588677, 9.077630899636648, 9.371600988533814, 10.14651399960436, 10.49217390902053, 11.01516890071478, 11.32205522591335, 11.53132242082182, 12.23848388696782, 12.67238897954841