L(s) = 1 | − 3.62e5i·3-s − 4.97e7i·5-s + 7.73e9·7-s − 3.69e10·9-s + 9.29e11i·11-s + 1.06e13i·13-s − 1.79e13·15-s − 2.44e14·17-s − 8.89e14i·19-s − 2.80e15i·21-s − 8.02e14·23-s + 9.44e15·25-s − 2.07e16i·27-s + 8.69e16i·29-s − 6.31e16·31-s + ⋯ |
L(s) = 1 | − 1.18i·3-s − 0.455i·5-s + 1.47·7-s − 0.392·9-s + 0.982i·11-s + 1.64i·13-s − 0.537·15-s − 1.72·17-s − 1.75i·19-s − 1.74i·21-s − 0.175·23-s + 0.792·25-s − 0.716i·27-s + 1.32i·29-s − 0.446·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(\approx\) |
\(2.556487596\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.556487596\) |
\(L(\frac{25}{2})\) |
|
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 | \( 1 + 3.62e5iT - 9.41e10T^{2} \) |
| 5 | \( 1 + 4.97e7iT - 1.19e16T^{2} \) |
| 7 | \( 1 - 7.73e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 9.29e11iT - 8.95e23T^{2} \) |
| 13 | \( 1 - 1.06e13iT - 4.17e25T^{2} \) |
| 17 | \( 1 + 2.44e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 8.89e14iT - 2.57e29T^{2} \) |
| 23 | \( 1 + 8.02e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 8.69e16iT - 4.31e33T^{2} \) |
| 31 | \( 1 + 6.31e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 9.15e17iT - 1.17e36T^{2} \) |
| 41 | \( 1 - 2.73e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 7.42e18iT - 3.71e37T^{2} \) |
| 47 | \( 1 + 2.26e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 2.93e18iT - 4.55e39T^{2} \) |
| 59 | \( 1 - 4.03e19iT - 5.36e40T^{2} \) |
| 61 | \( 1 - 2.25e20iT - 1.15e41T^{2} \) |
| 67 | \( 1 - 7.92e20iT - 9.99e41T^{2} \) |
| 71 | \( 1 - 3.47e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 4.69e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.80e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 2.09e21iT - 1.37e44T^{2} \) |
| 89 | \( 1 + 3.89e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 3.55e22T + 4.96e45T^{2} \) |
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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
−11.01266604373883913698582878977, −9.174466278711306157756996631341, −8.466350662122810650965934188935, −7.09570683633705664671281480535, −6.79272917996514012316518299758, −4.86621906929850857509921745848, −4.45951495388803011233845763117, −2.21793626390858248028775366436, −1.81433027628564288037369033376, −0.857418400700531752571983092969,
0.52586371239281499750474729527, 1.89631357069342688497828047470, 3.18130633609844586665147992860, 4.16380096682535369391239917442, 5.10161530017694173578689914378, 6.07841635495500821618796000206, 7.80948605759538274824992401200, 8.487999318399090305014326024013, 9.829628332210138190404045923899, 10.93248847272243668514472764056