L(s) = 1 | + (0.707 − 0.707i)3-s − 7-s − i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − 23-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s − 7-s − i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − 23-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07981111830 + 0.2027810986i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07981111830 + 0.2027810986i\) |
\(L(1)\) |
\(\approx\) |
\(0.8525966743 - 0.1153223352i\) |
\(L(1)\) |
\(\approx\) |
\(0.8525966743 - 0.1153223352i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.707 + 0.707i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.707 - 0.707i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.707 - 0.707i)T \) |
| 67 | \( 1 + (-0.707 + 0.707i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.707 + 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 - iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.00972916791462930949821280038, −26.32531053732845755749464608221, −25.37046998877987155430647117335, −24.577647420998334614833993940349, −23.09278704082799075451964984365, −22.2506420608529252562547043305, −21.29970105610677630641133663170, −20.33242701366020686449763109767, −19.43020908248412844892154695984, −18.58219029173022952672851272741, −17.0296194002522163493892431228, −16.00312055264628369194619061492, −15.40451114382543454720007666738, −14.07827020741281670653368058224, −13.2872211153691236372513186372, −12.03390716666866944228928613135, −10.405853244399514859220775417224, −9.901678084619627211908837742444, −8.61918531455775110826864475423, −7.64032432721772855233994811884, −6.06908410664594062037771826032, −4.78240766195343510451581709691, −3.386353355316583829596316506900, −2.52362158153559086012638082101, −0.066890065052554111909579382640,
1.90195784451822220503839948331, 2.99316453917975398799962947474, 4.41731248953125935014595507156, 6.23959132729753004492265230181, 7.08401167405166132094105823294, 8.25240995924161066413763558259, 9.390359037249696246388605697141, 10.36162736207258187583251678414, 12.14059223357293293452050151277, 12.7836546150922194643640382561, 13.76853012472557146032850779299, 14.887471025382976103316055706162, 15.8233573929068248377590275548, 17.1830058094368382858030632213, 18.184725529964373656718175886652, 19.35942886251866868238889424285, 19.72181325441686839396809855573, 21.00886480724344850296223031320, 22.05728171973624659583484410913, 23.36334066538258421423624676875, 23.98849091296132890140007520130, 25.1959587132099039680611108936, 26.017681649864817800649854145228, 26.51083105301822997359450854896, 28.19439225610654939347740276056