L(s) = 1 | − 3i·5-s + 3·7-s − 3i·11-s + 6i·13-s + 6·17-s + 2i·19-s + 6·23-s − 4·25-s + 6i·29-s + 3·31-s − 9i·35-s − 6i·37-s + 6·41-s − 8i·43-s − 12·47-s + ⋯ |
L(s) = 1 | − 1.34i·5-s + 1.13·7-s − 0.904i·11-s + 1.66i·13-s + 1.45·17-s + 0.458i·19-s + 1.25·23-s − 0.800·25-s + 1.11i·29-s + 0.538·31-s − 1.52i·35-s − 0.986i·37-s + 0.937·41-s − 1.21i·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.124225977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.124225977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−9.006012659423267900666190988806, −8.554463512971505208332575214289, −7.84129509268527927250600321243, −6.91242470636671259427802672516, −5.73870367187247599693109243408, −5.07860047042322202349512194047, −4.42236060216706181384132011387, −3.37911633933529852951966615371, −1.77025280910176116520663795636, −1.02738354571215150142269014578,
1.21409406908452432742353363030, 2.65157290845513525608423542709, 3.21165482118584790844721391666, 4.56051684217677194176409927739, 5.31701716934340813953930178113, 6.24333476053035838603584058915, 7.21341749490191112740640486581, 7.77572782486253416961356003743, 8.335705832750782116259770359924, 9.778167774539802117147448433113