L(s) = 1 | + (−0.00942 − 2.23i)5-s + (−1.55 + 1.55i)7-s − 6.24i·11-s + (−4.66 − 4.66i)13-s + (−2.07 − 2.07i)17-s − 3.70i·19-s + (−0.707 + 0.707i)23-s + (−4.99 + 0.0421i)25-s + 7.12·29-s − 2.07·31-s + (3.48 + 3.45i)35-s + (−1.56 + 1.56i)37-s + 0.905i·41-s + (5.14 + 5.14i)43-s + (1.35 + 1.35i)47-s + ⋯ |
L(s) = 1 | + (−0.00421 − 0.999i)5-s + (−0.587 + 0.587i)7-s − 1.88i·11-s + (−1.29 − 1.29i)13-s + (−0.503 − 0.503i)17-s − 0.849i·19-s + (−0.147 + 0.147i)23-s + (−0.999 + 0.00843i)25-s + 1.32·29-s − 0.373·31-s + (0.589 + 0.584i)35-s + (−0.257 + 0.257i)37-s + 0.141i·41-s + (0.784 + 0.784i)43-s + (0.198 + 0.198i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4670882002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4670882002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.00942 + 2.23i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (1.55 - 1.55i)T - 7iT^{2} \) |
| 11 | \( 1 + 6.24iT - 11T^{2} \) |
| 13 | \( 1 + (4.66 + 4.66i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.07 + 2.07i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.70iT - 19T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 2.07T + 31T^{2} \) |
| 37 | \( 1 + (1.56 - 1.56i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.905iT - 41T^{2} \) |
| 43 | \( 1 + (-5.14 - 5.14i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.35 - 1.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.81 - 7.81i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.74T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 + (3.97 - 3.97i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.59iT - 71T^{2} \) |
| 73 | \( 1 + (-5.20 - 5.20i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.96iT - 79T^{2} \) |
| 83 | \( 1 + (8.81 - 8.81i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 + (-5.50 + 5.50i)T - 97iT^{2} \) |
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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
−8.186369077540688423202430655473, −7.33389867543607548647131978745, −6.32709564092259087919273783303, −5.65111937716226246689287088294, −5.12216735429934195926557502688, −4.26507664682880425007743064782, −2.98841190357944181379276502370, −2.69255856710340732896760475511, −0.986211167561276637955583861006, −0.14751612461090710566695138769,
1.89948653252911384376517216932, 2.36243771976776403695650026781, 3.61161608240318353578796415958, 4.27761786334770850891568346317, 4.96750818054125995889051514295, 6.22232522078151567541807804965, 6.87688557601103940318631476872, 7.16167314879182201256554759240, 7.88275581330447390504278876269, 9.023978685675138933845707608853