L(s) = 1 | + 5.29·3-s − 10.5·5-s − 8·7-s + 1.00·9-s + 15.8·11-s − 52.9·13-s − 56.0·15-s − 14·17-s − 37.0·19-s − 42.3·21-s − 152·23-s − 12.9·25-s − 137.·27-s + 158.·29-s − 224·31-s + 84.0·33-s + 84.6·35-s + 243.·37-s − 280·39-s + 70·41-s + 439.·43-s − 10.5·45-s − 336·47-s − 279·49-s − 74.0·51-s + 31.7·53-s − 168.·55-s + ⋯ |
L(s) = 1 | + 1.01·3-s − 0.946·5-s − 0.431·7-s + 0.0370·9-s + 0.435·11-s − 1.12·13-s − 0.963·15-s − 0.199·17-s − 0.447·19-s − 0.439·21-s − 1.37·23-s − 0.103·25-s − 0.980·27-s + 1.01·29-s − 1.29·31-s + 0.443·33-s + 0.408·35-s + 1.08·37-s − 1.14·39-s + 0.266·41-s + 1.55·43-s − 0.0350·45-s − 1.04·47-s − 0.813·49-s − 0.203·51-s + 0.0822·53-s − 0.411·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 5.29T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 + 8T + 343T^{2} \) |
| 11 | \( 1 - 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 336T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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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
−11.20779462107548362820389961034, −9.904389925653341109409051794734, −9.115197621677218962708290685245, −8.084169857131592214349490346657, −7.45029228985168252547588112497, −6.13702733929959974461745957032, −4.45775212716799431414023613932, −3.48515460614466710686278551297, −2.28905405406886652526943167107, 0,
2.28905405406886652526943167107, 3.48515460614466710686278551297, 4.45775212716799431414023613932, 6.13702733929959974461745957032, 7.45029228985168252547588112497, 8.084169857131592214349490346657, 9.115197621677218962708290685245, 9.904389925653341109409051794734, 11.20779462107548362820389961034