L(s) = 1 | + (0.707 − 0.707i)3-s + (−2.12 − 1.57i)7-s − 1.00i·9-s − 2.40·11-s + (−0.697 + 0.697i)13-s + (−3.50 − 3.50i)17-s + 0.306·19-s + (−2.61 + 0.391i)21-s + (2.63 + 2.63i)23-s + (−0.707 − 0.707i)27-s + 4.12i·29-s + 9.14i·31-s + (−1.69 + 1.69i)33-s + (−6.11 + 6.11i)37-s + 0.986i·39-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.803 − 0.594i)7-s − 0.333i·9-s − 0.723·11-s + (−0.193 + 0.193i)13-s + (−0.849 − 0.849i)17-s + 0.0702·19-s + (−0.571 + 0.0853i)21-s + (0.548 + 0.548i)23-s + (−0.136 − 0.136i)27-s + 0.766i·29-s + 1.64i·31-s + (−0.295 + 0.295i)33-s + (−1.00 + 1.00i)37-s + 0.157i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2060639863\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2060639863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.12 + 1.57i)T \) |
good | 11 | \( 1 + 2.40T + 11T^{2} \) |
| 13 | \( 1 + (0.697 - 0.697i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.50 + 3.50i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.306T + 19T^{2} \) |
| 23 | \( 1 + (-2.63 - 2.63i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.12iT - 29T^{2} \) |
| 31 | \( 1 - 9.14iT - 31T^{2} \) |
| 37 | \( 1 + (6.11 - 6.11i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.06iT - 41T^{2} \) |
| 43 | \( 1 + (3.56 + 3.56i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.325 - 0.325i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.00 - 9.00i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 6.43iT - 61T^{2} \) |
| 67 | \( 1 + (9.48 - 9.48i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 + (3.69 - 3.69i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 + (-2.89 + 2.89i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 + (-7.96 - 7.96i)T + 97iT^{2} \) |
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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
−9.096812100892796564315730091029, −8.852116500252426439941533911495, −7.62778754276363437659891753375, −7.10113702584212228317327398741, −6.52234765678782662345618944015, −5.36605508020760621938657184218, −4.55122232454912835753869713201, −3.35670909340788451233918165764, −2.77367431263989572301717911605, −1.43444032288825731691121828375,
0.06512367502982419525269247096, 2.10848735652561526438286269385, 2.84581999664207717470894236790, 3.82622271398925373812008863243, 4.71957633301082313492077075103, 5.69525201078325128646023627697, 6.35923907111691182465349435453, 7.35705195327417637029910129120, 8.189123908295529424054918944671, 8.870224241996317831010634211931