L(s) = 1 | − 3.70i·2-s − 5.75·4-s − 4.54·5-s − 8.32i·8-s + 16.8i·10-s + 45.5i·11-s + 11.6i·13-s − 76.9·16-s + 91.5·17-s + 140. i·19-s + 26.1·20-s + 168.·22-s − 45.0i·23-s − 104.·25-s + 43.3·26-s + ⋯ |
L(s) = 1 | − 1.31i·2-s − 0.719·4-s − 0.406·5-s − 0.367i·8-s + 0.533i·10-s + 1.24i·11-s + 0.249i·13-s − 1.20·16-s + 1.30·17-s + 1.69i·19-s + 0.292·20-s + 1.63·22-s − 0.408i·23-s − 0.834·25-s + 0.326·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.555201482\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555201482\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 3.70iT - 8T^{2} \) |
| 5 | \( 1 + 4.54T + 125T^{2} \) |
| 11 | \( 1 - 45.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 11.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 91.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 45.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 47.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 238. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 148.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 412.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 167. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 323. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 424.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 727. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 669.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 199.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 807.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 701. iT - 9.12e5T^{2} \) |
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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
−10.53382197201612418368801334083, −10.05933282042795630862471827210, −9.211713692532856173724779125323, −7.919952544395602124782301538173, −7.10033419134049246091404158430, −5.73889203718845113140387293457, −4.33540463587911142058634880045, −3.58956764561232321425669010067, −2.28279337922907540290627459152, −1.18530496253690892859891729779,
0.57234086036015337844726313961, 2.74475262805260924571303235058, 4.10325832534192508375587544023, 5.45318228843607339017310498826, 5.97858191218867114809186182172, 7.21155671550294662587341963705, 7.80210636268854311227004226891, 8.671706723640067121468024947382, 9.535665444692669164688436005892, 10.97320938166859654754838598039