| L(s) = 1 | + 1.41i·5-s − 10·13-s − 9.89i·17-s + 23·25-s + 1.41i·29-s + 24·37-s + 43.8i·41-s − 49·49-s + 103. i·53-s − 120·61-s − 14.1i·65-s + 96·73-s + 14.0·85-s + 57.9i·89-s − 144·97-s + ⋯ |
| L(s) = 1 | + 0.282i·5-s − 0.769·13-s − 0.582i·17-s + 0.920·25-s + 0.0487i·29-s + 0.648·37-s + 1.06i·41-s − 0.999·49-s + 1.94i·53-s − 1.96·61-s − 0.217i·65-s + 1.31·73-s + 0.164·85-s + 0.651i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9549847279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9549847279\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.41iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 9.89iT - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 1.41iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 24T + 1.36e3T^{2} \) |
| 41 | \( 1 - 43.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 103. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 120T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−9.270542100350163152045461730676, −8.233438140587253853821981784961, −7.54221127866029483346268421478, −6.81310507857374958286761146438, −6.04527932824353888194720510008, −5.03752340420401594624228253318, −4.40084802629761769561590640970, −3.17896638730828637839780350686, −2.48954405627249435765450686840, −1.16093078866664605686047485253,
0.24240646998371592910354209018, 1.56324985060688860019196027432, 2.63081903704421381613794788898, 3.65359016581579352616013211268, 4.62783806926802644101261378461, 5.29665539387993957677737498042, 6.25637194517092908262329306193, 7.01439924277547238011746504690, 7.84813023267881509331664209672, 8.540367627264066963812628938294
