L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.86 − 1.22i)5-s + (0.329 − 0.329i)7-s − 1.00i·9-s − 2.84i·11-s + (−3.80 + 3.80i)13-s + (−2.18 + 0.453i)15-s + (−3.63 + 3.63i)17-s − 3.13·19-s − 0.465i·21-s + (−4.79 + 0.0646i)23-s + (1.98 + 4.58i)25-s + (−0.707 − 0.707i)27-s + 2.87i·29-s + 4.47·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.835 − 0.548i)5-s + (0.124 − 0.124i)7-s − 0.333i·9-s − 0.857i·11-s + (−1.05 + 1.05i)13-s + (−0.565 + 0.117i)15-s + (−0.882 + 0.882i)17-s − 0.718·19-s − 0.101i·21-s + (−0.999 + 0.0134i)23-s + (0.397 + 0.917i)25-s + (−0.136 − 0.136i)27-s + 0.534i·29-s + 0.804·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09321391717\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09321391717\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.86 + 1.22i)T \) |
| 23 | \( 1 + (4.79 - 0.0646i)T \) |
good | 7 | \( 1 + (-0.329 + 0.329i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.84iT - 11T^{2} \) |
| 13 | \( 1 + (3.80 - 3.80i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.63 - 3.63i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 29 | \( 1 - 2.87iT - 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + (-2.51 + 2.51i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + (-1.58 - 1.58i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.81 - 1.81i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.43 + 3.43i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 1.84iT - 61T^{2} \) |
| 67 | \( 1 + (10.9 - 10.9i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + (6.01 - 6.01i)T - 73iT^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + (-5.51 - 5.51i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + (-7.16 + 7.16i)T - 97iT^{2} \) |
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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
−9.703368518150967531286054509326, −8.827587405909411108316131186261, −8.348335403722186022810636317869, −7.56444103969250904231835965270, −6.73546787670613026164381352431, −5.85953226926198202756090127387, −4.49989557633087727646181680527, −4.07240333214402457419197096898, −2.76291550758772412033975809634, −1.58006726922218032735093133579,
0.03419225320698125777211154002, 2.29504588324592652919723281121, 2.98581864764003541643564216814, 4.28601777127368033567040872126, 4.70438089023523519845408884880, 5.99287354709253357007955138361, 7.09508772462851652250281512846, 7.65477746959770641712419747203, 8.383749518293380639601379116801, 9.322937064372856456363266815619