L(s) = 1 | − 2-s + (−1.23 + 1.23i)3-s + 4-s + (−1.26 + 1.84i)5-s + (1.23 − 1.23i)6-s + (1.28 − 1.28i)7-s − 8-s − 0.0591i·9-s + (1.26 − 1.84i)10-s + 4.27i·11-s + (−1.23 + 1.23i)12-s − 1.50·13-s + (−1.28 + 1.28i)14-s + (−0.708 − 3.84i)15-s + 16-s − 2.12i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.714 + 0.714i)3-s + 0.5·4-s + (−0.567 + 0.823i)5-s + (0.504 − 0.504i)6-s + (0.486 − 0.486i)7-s − 0.353·8-s − 0.0197i·9-s + (0.401 − 0.582i)10-s + 1.28i·11-s + (−0.357 + 0.357i)12-s − 0.418·13-s + (−0.343 + 0.343i)14-s + (−0.182 − 0.993i)15-s + 0.250·16-s − 0.514i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0159008 - 0.330015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0159008 - 0.330015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (1.26 - 1.84i)T \) |
| 37 | \( 1 + (5.63 - 2.28i)T \) |
good | 3 | \( 1 + (1.23 - 1.23i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.28 + 1.28i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.27iT - 11T^{2} \) |
| 13 | \( 1 + 1.50T + 13T^{2} \) |
| 17 | \( 1 + 2.12iT - 17T^{2} \) |
| 19 | \( 1 + (0.381 + 0.381i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 + (-0.279 + 0.279i)T - 29iT^{2} \) |
| 31 | \( 1 + (1.59 + 1.59i)T + 31iT^{2} \) |
| 41 | \( 1 + 0.270iT - 41T^{2} \) |
| 43 | \( 1 - 0.302T + 43T^{2} \) |
| 47 | \( 1 + (3.17 - 3.17i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.71 - 9.71i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.06 + 4.06i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.47 - 4.47i)T + 61iT^{2} \) |
| 67 | \( 1 + (6.17 + 6.17i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 73 | \( 1 + (-8.32 + 8.32i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.20 + 2.20i)T + 79iT^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.24 + 5.24i)T - 89iT^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−11.71344274319934092355729631513, −10.64040976016217645111400803766, −10.30796308701378352696202550950, −9.424647880610435644171278078648, −7.88212510254660794079181138333, −7.40738701223573808004123090406, −6.27839840305695842593853238188, −4.89518994682999238630700402437, −3.98511309186099783874256001543, −2.22615220952263096630196242262,
0.29745285762675828136349118838, 1.75549983936233523907562518504, 3.72680851803897255429906924026, 5.36112198198438168845403131649, 6.08834691295430008122030074110, 7.27366994226183147917517545482, 8.299194325665426695882212311653, 8.737563948130830268558952456921, 10.02219003187321127337467000362, 11.18258709526931217031690269929