L(s) = 1 | + (3.74 + 3.74i)2-s + 20.0i·4-s + (−1.80 + 1.80i)7-s + (−45.3 + 45.3i)8-s + 46.0i·11-s + (−18.6 − 18.6i)13-s − 13.5·14-s − 178.·16-s + (−14.5 − 14.5i)17-s + 74.4i·19-s + (−172. + 172. i)22-s + (59.6 − 59.6i)23-s − 139. i·26-s + (−36.3 − 36.3i)28-s + 202.·29-s + ⋯ |
L(s) = 1 | + (1.32 + 1.32i)2-s + 2.51i·4-s + (−0.0977 + 0.0977i)7-s + (−2.00 + 2.00i)8-s + 1.26i·11-s + (−0.397 − 0.397i)13-s − 0.258·14-s − 2.79·16-s + (−0.208 − 0.208i)17-s + 0.899i·19-s + (−1.67 + 1.67i)22-s + (0.540 − 0.540i)23-s − 1.05i·26-s + (−0.245 − 0.245i)28-s + 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0962923 + 3.10996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0962923 + 3.10996i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-3.74 - 3.74i)T + 8iT^{2} \) |
| 7 | \( 1 + (1.80 - 1.80i)T - 343iT^{2} \) |
| 11 | \( 1 - 46.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (18.6 + 18.6i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (14.5 + 14.5i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 74.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-59.6 + 59.6i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 49.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-45.0 + 45.0i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 306. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-230. - 230. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-176. - 176. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-85.1 + 85.1i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 330.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 678.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (756. - 756. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 100. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (586. + 586. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 286. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-947. + 947. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 688.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (920. - 920. i)T - 9.12e5iT^{2} \) |
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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
−12.52120272023255630301232710219, −11.93965300129028314111124312926, −10.35620743800867504114469407524, −8.991306523191797285780427236766, −7.80639942030658494603589618942, −7.08484869311437010274597198754, −6.06973480499400735093525933959, −4.99358141958485260258351442908, −4.12904658527716359467371159150, −2.67227920963814486856359467335,
0.855123952330764660102799670898, 2.48963329746032895675964270330, 3.53840430114179321270488573225, 4.69584095232736864395392268517, 5.71426789104380861367450085462, 6.81531419982414648806161505109, 8.687719699888572171143851848699, 9.776552956449800098361139107977, 10.78707091365051295929412323313, 11.42580120790952588182627574520