L(s) = 1 | + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (2.04 + 0.914i)5-s + (−0.707 − 0.707i)6-s + (0.867 − 0.867i)7-s − i·8-s − 1.00i·9-s + (−0.914 + 2.04i)10-s − 1.09i·11-s + (0.707 − 0.707i)12-s − 4.92i·13-s + (0.867 + 0.867i)14-s + (−2.08 + 0.796i)15-s + 16-s + 2.47·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.912 + 0.408i)5-s + (−0.288 − 0.288i)6-s + (0.327 − 0.327i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.289 + 0.645i)10-s − 0.331i·11-s + (0.204 − 0.204i)12-s − 1.36i·13-s + (0.231 + 0.231i)14-s + (−0.539 + 0.205i)15-s + 0.250·16-s + 0.600·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.631760624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.631760624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.04 - 0.914i)T \) |
| 37 | \( 1 + (6.08 + 0.134i)T \) |
good | 7 | \( 1 + (-0.867 + 0.867i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.09iT - 11T^{2} \) |
| 13 | \( 1 + 4.92iT - 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + (-5.01 + 5.01i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.254iT - 23T^{2} \) |
| 29 | \( 1 + (4.52 + 4.52i)T + 29iT^{2} \) |
| 31 | \( 1 + (-3.38 + 3.38i)T - 31iT^{2} \) |
| 41 | \( 1 - 5.96iT - 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 + (-2.51 + 2.51i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.28 + 1.28i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.61 - 3.61i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.890 + 0.890i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.08 - 8.08i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + (9.56 - 9.56i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.43 + 1.43i)T - 79iT^{2} \) |
| 83 | \( 1 + (-7.34 - 7.34i)T + 83iT^{2} \) |
| 89 | \( 1 + (-9.93 - 9.93i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.50T + 97T^{2} \) |
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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
−9.870359086152409630143658968989, −9.218762653568546233889322347669, −8.134229830050231569674596761169, −7.35023903976858393216937102913, −6.47695459771593493042544554176, −5.45505256936906503214587274727, −5.24644943211059495881825501564, −3.81644489079840192832868942253, −2.73851547152904764468940300335, −0.870045787899632228502984165871,
1.37953356861540435761320571130, 1.98242783318542237214481451473, 3.40375388593140823962829893841, 4.72568434603564929160969922042, 5.38723441193691558239792649661, 6.25824735726924421021185051387, 7.28107640486027908661347368221, 8.292649144098144810457222327723, 9.214385814029960479343879276567, 9.746164047321307285076863041419