L(s) = 1 | − 2.97i·3-s − 2.24i·5-s − 1.80i·7-s − 5.86·9-s − 3.56i·11-s − 3.28·13-s − 6.68·15-s + (−0.415 − 4.10i)17-s − 3.16·19-s − 5.37·21-s − 3.51i·23-s − 0.0370·25-s + 8.51i·27-s − 4.99i·29-s − 6.66i·31-s + ⋯ |
L(s) = 1 | − 1.71i·3-s − 1.00i·5-s − 0.682i·7-s − 1.95·9-s − 1.07i·11-s − 0.910·13-s − 1.72·15-s + (−0.100 − 0.994i)17-s − 0.725·19-s − 1.17·21-s − 0.732i·23-s − 0.00741·25-s + 1.63i·27-s − 0.927i·29-s − 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.385807115\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385807115\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (0.415 + 4.10i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 2.97iT - 3T^{2} \) |
| 5 | \( 1 + 2.24iT - 5T^{2} \) |
| 7 | \( 1 + 1.80iT - 7T^{2} \) |
| 11 | \( 1 + 3.56iT - 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + 3.51iT - 23T^{2} \) |
| 29 | \( 1 + 4.99iT - 29T^{2} \) |
| 31 | \( 1 + 6.66iT - 31T^{2} \) |
| 37 | \( 1 - 4.52iT - 37T^{2} \) |
| 41 | \( 1 - 7.64iT - 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 0.459T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 61 | \( 1 + 4.46iT - 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 0.964iT - 71T^{2} \) |
| 73 | \( 1 + 1.39iT - 73T^{2} \) |
| 79 | \( 1 + 15.9iT - 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 4.70T + 89T^{2} \) |
| 97 | \( 1 - 9.04iT - 97T^{2} \) |
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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
−7.81380908342040324612366064751, −7.32208338272562287097738540557, −6.45836062493965641658112876829, −5.92819543384598717953091242685, −4.96854394853000711388259178762, −4.18687839220557725044703323619, −2.83853796550642349826049868568, −2.15043392606386815461694698236, −0.845232524239757691835817273924, −0.51162114719795465808903759892,
2.11794966572839830852937491567, 2.81374913577091969731110844689, 3.75585056399799851762836563189, 4.34276657326797209851948402259, 5.24173431091863301790242060772, 5.74934597523041560713622970629, 6.83493867749241504299793679110, 7.41428636534073767276786062923, 8.602980260075299190392101092767, 9.039449457630130382459518042071