L(s) = 1 | + (−1.73 − 1.73i)2-s + 3.99i·4-s + (1.22 − 1.22i)7-s + (3.46 − 3.46i)8-s − 4.24i·11-s + (−3.67 − 3.67i)13-s − 4.24·14-s − 3.99·16-s + (1.73 + 1.73i)17-s − 5i·19-s + (−7.34 + 7.34i)22-s + (−1.73 + 1.73i)23-s + 12.7i·26-s + (4.89 + 4.89i)28-s − 4.24·29-s + ⋯ |
L(s) = 1 | + (−1.22 − 1.22i)2-s + 1.99i·4-s + (0.462 − 0.462i)7-s + (1.22 − 1.22i)8-s − 1.27i·11-s + (−1.01 − 1.01i)13-s − 1.13·14-s − 0.999·16-s + (0.420 + 0.420i)17-s − 1.14i·19-s + (−1.56 + 1.56i)22-s + (−0.361 + 0.361i)23-s + 2.49i·26-s + (0.925 + 0.925i)28-s − 0.787·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.142229 - 0.554215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.142229 - 0.554215i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.73 + 1.73i)T + 2iT^{2} \) |
| 7 | \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (3.67 + 3.67i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.73 - 1.73i)T + 17iT^{2} \) |
| 19 | \( 1 + 5iT - 19T^{2} \) |
| 23 | \( 1 + (1.73 - 1.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.19 - 5.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.92 + 6.92i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (1.73 - 1.73i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + (8.57 - 8.57i)T - 97iT^{2} \) |
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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
−11.47590850967784301914375074342, −10.82830197752965162340494667071, −10.06358246872476802783446505034, −9.046415633833492609780552262527, −8.137131401640558919117113142037, −7.34616539281897897653365323572, −5.48793683036491656874411270155, −3.73322246618618312885169621328, −2.51179118645682904468238432214, −0.71881077159523051674139860729,
1.91646681515026274660386773448, 4.59764290092272401539857748534, 5.73695267404867297466864619600, 6.94932558777755585588789359843, 7.62843142538659275611409059202, 8.609706231969304663220920502309, 9.663364362850317436610656346382, 10.08642708235202175042730428718, 11.62457466386676957517898233086, 12.46124525936878305766155436051