L(s) = 1 | + (1.22 + 1.22i)3-s + (−2.44 + 2.44i)7-s + 2.99i·9-s + (4.89 + 4.89i)13-s − 8i·19-s − 5.99·21-s + (−3.67 + 3.67i)27-s + 4·31-s + (4.89 − 4.89i)37-s + 11.9i·39-s + (−7.34 − 7.34i)43-s − 4.99i·49-s + (9.79 − 9.79i)57-s + 14·61-s + (−7.34 − 7.34i)63-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.925 + 0.925i)7-s + 0.999i·9-s + (1.35 + 1.35i)13-s − 1.83i·19-s − 1.30·21-s + (−0.707 + 0.707i)27-s + 0.718·31-s + (0.805 − 0.805i)37-s + 1.92i·39-s + (−1.12 − 1.12i)43-s − 0.714i·49-s + (1.29 − 1.29i)57-s + 1.79·61-s + (−0.925 − 0.925i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17066 + 0.926483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17066 + 0.926483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.44 - 2.44i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.89 - 4.89i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (7.34 + 7.34i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (2.44 - 2.44i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (9.79 + 9.79i)T + 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-9.79 + 9.79i)T - 97iT^{2} \) |
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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
−11.77429025792328301980138238785, −11.01320479877194739921987132597, −9.834070822083326400729409964486, −8.998034393088949535561727581637, −8.628485723260140079931619149430, −7.03338182723814703896016506530, −6.02158778135602199514818276050, −4.67163413279996298194903414777, −3.51011191800166728322480057009, −2.34826722357148605880232686547,
1.14306251508691973050411521430, 3.10809987766291724972668468928, 3.87033357540568871028019178587, 5.89180492702692051383416284098, 6.65378478733536299864913578006, 7.86339715024447441911602339562, 8.395828040338956258713311183158, 9.794838228001785096407611600209, 10.37110439937222862078897240559, 11.66766425637560411303034890671