L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−0.461 − 2.18i)5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−1.22 + 1.87i)10-s − 2.98·11-s + (−0.707 + 0.707i)12-s + (0.960 + 0.960i)13-s + (1.22 − 1.87i)15-s − 1.00·16-s + (1.62 − 1.62i)17-s + (0.707 − 0.707i)18-s − 8.67·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.206 − 0.978i)5-s − 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.385 + 0.592i)10-s − 0.899·11-s + (−0.204 + 0.204i)12-s + (0.266 + 0.266i)13-s + (0.315 − 0.483i)15-s − 0.250·16-s + (0.394 − 0.394i)17-s + (0.166 − 0.166i)18-s − 1.98·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1386217028\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1386217028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.461 + 2.18i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 + (-0.960 - 0.960i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.62 + 1.62i)T - 17iT^{2} \) |
| 19 | \( 1 + 8.67T + 19T^{2} \) |
| 23 | \( 1 + (-1.36 + 1.36i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 + 0.179iT - 31T^{2} \) |
| 37 | \( 1 + (4.86 + 4.86i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.14iT - 41T^{2} \) |
| 43 | \( 1 + (7.01 - 7.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.202 - 0.202i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.01 - 7.01i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 2.41iT - 61T^{2} \) |
| 67 | \( 1 + (6.29 + 6.29i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 + (-8.78 - 8.78i)T + 73iT^{2} \) |
| 79 | \( 1 - 16.2iT - 79T^{2} \) |
| 83 | \( 1 + (8.11 + 8.11i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + (7.44 - 7.44i)T - 97iT^{2} \) |
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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
−8.996449761736337497130879250275, −8.438379956767564269489099598940, −7.85904211003159614839143099838, −6.79045169184473963358948514827, −5.53148723441909341649519668926, −4.64251471139207609541391336669, −3.89758836099578182421450855807, −2.74297349712344090262764238963, −1.66678970371135393263578258134, −0.05811793773772736151354023646,
1.83425534472298242389152163744, 2.84394170236271302641391484750, 3.88259481332010388989377564983, 5.16245872423011260811563208424, 6.26176753196388320122660504128, 6.71759176832934356472460780643, 7.69734205579460001320521692601, 8.187382319304747166729204159592, 8.900132496480282363809283908701, 10.14073231192835204073120163295