L(s) = 1 | + (−1.23 − 0.683i)2-s + (0.494 − 0.494i)3-s + (1.06 + 1.69i)4-s + (1.31 + 1.81i)5-s + (−0.950 + 0.274i)6-s + (−0.327 − 0.327i)7-s + (−0.165 − 2.82i)8-s + 2.51i·9-s + (−0.385 − 3.13i)10-s + 2.94i·11-s + (1.36 + 0.309i)12-s + (0.287 + 0.287i)13-s + (0.181 + 0.629i)14-s + (1.54 + 0.247i)15-s + (−1.72 + 3.60i)16-s + (−1.74 + 1.74i)17-s + ⋯ |
L(s) = 1 | + (−0.875 − 0.483i)2-s + (0.285 − 0.285i)3-s + (0.533 + 0.845i)4-s + (0.586 + 0.810i)5-s + (−0.387 + 0.112i)6-s + (−0.123 − 0.123i)7-s + (−0.0584 − 0.998i)8-s + 0.836i·9-s + (−0.121 − 0.992i)10-s + 0.888i·11-s + (0.393 + 0.0892i)12-s + (0.0797 + 0.0797i)13-s + (0.0486 + 0.168i)14-s + (0.398 + 0.0639i)15-s + (−0.430 + 0.902i)16-s + (−0.422 + 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.971871 + 0.319592i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.971871 + 0.319592i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.23 + 0.683i)T \) |
| 5 | \( 1 + (-1.31 - 1.81i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (-0.494 + 0.494i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.327 + 0.327i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.94iT - 11T^{2} \) |
| 13 | \( 1 + (-0.287 - 0.287i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.74 - 1.74i)T - 17iT^{2} \) |
| 23 | \( 1 + (0.570 - 0.570i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.85iT - 29T^{2} \) |
| 31 | \( 1 + 1.12iT - 31T^{2} \) |
| 37 | \( 1 + (-4.08 + 4.08i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + (-7.09 + 7.09i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.57 - 2.57i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.67 - 5.67i)T + 53iT^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.69T + 61T^{2} \) |
| 67 | \( 1 + (9.51 + 9.51i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.19iT - 71T^{2} \) |
| 73 | \( 1 + (4.82 + 4.82i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.164T + 79T^{2} \) |
| 83 | \( 1 + (1.98 - 1.98i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.27iT - 89T^{2} \) |
| 97 | \( 1 + (4.13 - 4.13i)T - 97iT^{2} \) |
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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
−11.02284130187919145750859206424, −10.59867721644167306546589051763, −9.681647893380046523314308855574, −8.837482474326336573138609673023, −7.62824807820011470679703317338, −7.12065255058908648513554363576, −5.95248408567350137497768455766, −4.19320748040702036028544966947, −2.69374992504772740845882181189, −1.86285167920877983778765682720,
0.912744270929532088383371867880, 2.68254517170523021301270248069, 4.43849178619570342182450876418, 5.78944410080198394295519258723, 6.37365908448799579985609671182, 7.76770905269166483543699535607, 8.771643060661518646740020742416, 9.207499151416508815679307850576, 10.01171099302248532336126352063, 11.04884318906777686225525645719