L(s) = 1 | − 1.21i·2-s + i·3-s + 0.525·4-s + (0.311 + 2.21i)5-s + 1.21·6-s + 4.90i·7-s − 3.06i·8-s − 9-s + (2.68 − 0.377i)10-s − 11-s + 0.525i·12-s − 4.14i·13-s + 5.95·14-s + (−2.21 + 0.311i)15-s − 2.67·16-s − 5.33i·17-s + ⋯ |
L(s) = 1 | − 0.858i·2-s + 0.577i·3-s + 0.262·4-s + (0.139 + 0.990i)5-s + 0.495·6-s + 1.85i·7-s − 1.08i·8-s − 0.333·9-s + (0.850 − 0.119i)10-s − 0.301·11-s + 0.151i·12-s − 1.15i·13-s + 1.59·14-s + (−0.571 + 0.0803i)15-s − 0.668·16-s − 1.29i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27678 + 0.0892547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27678 + 0.0892547i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.311 - 2.21i)T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.21iT - 2T^{2} \) |
| 7 | \( 1 - 4.90iT - 7T^{2} \) |
| 13 | \( 1 + 4.14iT - 13T^{2} \) |
| 17 | \( 1 + 5.33iT - 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 5.80iT - 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 4.90iT - 43T^{2} \) |
| 47 | \( 1 - 7.05iT - 47T^{2} \) |
| 53 | \( 1 - 7.18iT - 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 + 4.85iT - 67T^{2} \) |
| 71 | \( 1 - 0.428T + 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 - 2.90iT - 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 + 2.75iT - 97T^{2} \) |
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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
−12.42591700326772259135131961979, −11.75274499451468769814497832711, −10.93673614159179793001807129298, −9.987531896765628996007532085677, −9.213291890312310662632329746362, −7.72235765685648655468329526974, −6.27550311638454957668389070865, −5.22825250951456454655985277584, −3.08580578737002228461828351961, −2.60156475798082844688044219292,
1.52248668343682865175322114389, 4.02601372739391660383617864153, 5.41566941104883907629370461008, 6.66430025550253030416712953255, 7.49660902549236248131626906629, 8.263296230260132690868536655374, 9.647003665823371858094714042611, 10.92134474258131848440946901879, 11.85165422693635068718812965093, 13.12120820285906345804783116867