L(s) = 1 | + 1.43i·2-s + (−1.43 + 1.43i)3-s − 0.0521·4-s + 4.15i·5-s + (−2.04 − 2.04i)6-s + (0.707 − 0.707i)7-s + 2.79i·8-s − 1.09i·9-s − 5.94·10-s + (4.11 − 4.11i)11-s + (0.0746 − 0.0746i)12-s + (0.0880 − 0.0880i)13-s + (1.01 + 1.01i)14-s + (−5.94 − 5.94i)15-s − 4.10·16-s + (−2.13 − 2.13i)17-s + ⋯ |
L(s) = 1 | + 1.01i·2-s + (−0.825 + 0.825i)3-s − 0.0260·4-s + 1.85i·5-s + (−0.836 − 0.836i)6-s + (0.267 − 0.267i)7-s + 0.986i·8-s − 0.364i·9-s − 1.88·10-s + (1.23 − 1.23i)11-s + (0.0215 − 0.0215i)12-s + (0.0244 − 0.0244i)13-s + (0.270 + 0.270i)14-s + (−1.53 − 1.53i)15-s − 1.02·16-s + (−0.518 − 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0471315 - 1.13611i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0471315 - 1.13611i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (-5.98 - 2.28i)T \) |
good | 2 | \( 1 - 1.43iT - 2T^{2} \) |
| 3 | \( 1 + (1.43 - 1.43i)T - 3iT^{2} \) |
| 5 | \( 1 - 4.15iT - 5T^{2} \) |
| 11 | \( 1 + (-4.11 + 4.11i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.0880 + 0.0880i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.13 + 2.13i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.297 - 0.297i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 + (-5.55 + 5.55i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.142T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 43 | \( 1 - 3.03iT - 43T^{2} \) |
| 47 | \( 1 + (-3.23 - 3.23i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.33 - 2.33i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-0.401 - 0.401i)T + 67iT^{2} \) |
| 71 | \( 1 + (9.25 - 9.25i)T - 71iT^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + (-4.00 + 4.00i)T - 79iT^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + (10.4 - 10.4i)T - 89iT^{2} \) |
| 97 | \( 1 + (5.89 + 5.89i)T + 97iT^{2} \) |
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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
−11.64807363258470289021725486634, −11.20809663238692979042181363229, −10.74498147492133777211990428569, −9.597675339233132671367449782041, −8.214226212908261128415196265668, −7.06593400070369895262054378914, −6.40714951733508707115010709099, −5.67944697707204987805147083459, −4.23632651675337832072970268213, −2.83691221821338856206742195996,
1.04924076853259506580118537254, 1.81715522153348292320295101128, 4.07324632679035057793918922125, 5.05411707534521169524214743384, 6.35588218593591003774753308173, 7.29133183516281231019953524346, 8.828689823172096693475028140722, 9.379023658359827578585209618291, 10.67901488963885642899148807081, 11.89613333106329586886180808632