L(s) = 1 | − 3.23·2-s + (−3.88 − 3.45i)3-s + 2.45·4-s + (8.12 + 7.68i)5-s + (12.5 + 11.1i)6-s + (−17.9 + 4.72i)7-s + 17.9·8-s + (3.15 + 26.8i)9-s + (−26.2 − 24.8i)10-s − 0.605i·11-s + (−9.52 − 8.47i)12-s + 12.8·13-s + (57.8 − 15.2i)14-s + (−5.01 − 57.8i)15-s − 77.6·16-s − 117. i·17-s + ⋯ |
L(s) = 1 | − 1.14·2-s + (−0.747 − 0.664i)3-s + 0.306·4-s + (0.726 + 0.687i)5-s + (0.854 + 0.759i)6-s + (−0.966 + 0.255i)7-s + 0.792·8-s + (0.116 + 0.993i)9-s + (−0.830 − 0.785i)10-s − 0.0165i·11-s + (−0.229 − 0.203i)12-s + 0.273·13-s + (1.10 − 0.291i)14-s + (−0.0863 − 0.996i)15-s − 1.21·16-s − 1.68i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.441314 - 0.310505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.441314 - 0.310505i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.88 + 3.45i)T \) |
| 5 | \( 1 + (-8.12 - 7.68i)T \) |
| 7 | \( 1 + (17.9 - 4.72i)T \) |
good | 2 | \( 1 + 3.23T + 8T^{2} \) |
| 11 | \( 1 + 0.605iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 98.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 131. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 58.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 519. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 104. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 172. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 622. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 151. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 94.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 779. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−13.37605926857797786381973381014, −11.73186001865107971307400087236, −10.78336495230091597080307234577, −9.802469580557588945086018660963, −8.946052430725478079950062807650, −7.27857978401233533809263436843, −6.66727474577641913317034562314, −5.21111326750951112619503305081, −2.53334649492707021037428349552, −0.58850200650362006242532512497,
1.16069636495234623566547990001, 3.98602873239404821150266657975, 5.52502966700550208734375265750, 6.68782165123894361454221967647, 8.443659552372258811453288853661, 9.341545792922385000355442617796, 10.16090436637261267623757341079, 10.79445219953570299922683500978, 12.47638748383200506689818141735, 13.19345151478423614049557235928