| L(s) = 1 | + 5.13·5-s + 4.02i·7-s + 4.30i·11-s − 18.4·13-s − 23.5·17-s + 21.7i·19-s + 30.7i·23-s + 1.34·25-s − 12.6·29-s − 24.5i·31-s + 20.6i·35-s + 18.2·37-s − 38.0·41-s − 34.9i·43-s − 29.6i·47-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.575i·7-s + 0.391i·11-s − 1.42·13-s − 1.38·17-s + 1.14i·19-s + 1.33i·23-s + 0.0536·25-s − 0.437·29-s − 0.791i·31-s + 0.590i·35-s + 0.492·37-s − 0.929·41-s − 0.813i·43-s − 0.630i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.079782148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079782148\) |
| \(L(2)\) |
|
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 | \( 1 - 5.13T + 25T^{2} \) |
| 7 | \( 1 - 4.02iT - 49T^{2} \) |
| 11 | \( 1 - 4.30iT - 121T^{2} \) |
| 13 | \( 1 + 18.4T + 169T^{2} \) |
| 17 | \( 1 + 23.5T + 289T^{2} \) |
| 19 | \( 1 - 21.7iT - 361T^{2} \) |
| 23 | \( 1 - 30.7iT - 529T^{2} \) |
| 29 | \( 1 + 12.6T + 841T^{2} \) |
| 31 | \( 1 + 24.5iT - 961T^{2} \) |
| 37 | \( 1 - 18.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 34.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 29.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 65.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 29.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 11.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 140. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 119.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 9.18iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.88T + 7.92e3T^{2} \) |
| 97 | \( 1 - 55.5T + 9.40e3T^{2} \) |
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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
−9.917156854931533901493900220363, −9.673194548598037493507858276312, −8.762356129585901202480187659556, −7.68120528397975616350883588144, −6.82134128686700171432226389680, −5.79851079484728005820988072871, −5.19249724687226245866341991459, −4.00655023697578928558186970297, −2.48828618689559473000765149978, −1.82742473779440736304159085626,
0.31674358294651405334881898116, 1.98737975734990628337206339338, 2.88177125807138898549192342283, 4.45178133347758899977961341294, 5.08462067032860526111778781960, 6.35439286715931492289093826768, 6.87624165856754820297351036100, 7.943008780607658678008410314708, 9.044435753741510479694631512681, 9.572847290290593912337523060356
