L(s) = 1 | + (−0.161 + 0.986i)2-s + (0.267 − 0.963i)3-s + (−0.947 − 0.319i)4-s + (−0.561 − 0.827i)5-s + (0.907 + 0.419i)6-s + (0.976 − 0.214i)7-s + (0.468 − 0.883i)8-s + (−0.856 − 0.515i)9-s + (0.907 − 0.419i)10-s + (0.796 − 0.605i)11-s + (−0.561 + 0.827i)12-s + (−0.856 + 0.515i)13-s + (0.0541 + 0.998i)14-s + (−0.947 + 0.319i)15-s + (0.796 + 0.605i)16-s + (0.976 + 0.214i)17-s + ⋯ |
L(s) = 1 | + (−0.161 + 0.986i)2-s + (0.267 − 0.963i)3-s + (−0.947 − 0.319i)4-s + (−0.561 − 0.827i)5-s + (0.907 + 0.419i)6-s + (0.976 − 0.214i)7-s + (0.468 − 0.883i)8-s + (−0.856 − 0.515i)9-s + (0.907 − 0.419i)10-s + (0.796 − 0.605i)11-s + (−0.561 + 0.827i)12-s + (−0.856 + 0.515i)13-s + (0.0541 + 0.998i)14-s + (−0.947 + 0.319i)15-s + (0.796 + 0.605i)16-s + (0.976 + 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7788700678 - 0.1771731835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7788700678 - 0.1771731835i\) |
\(L(1)\) |
\(\approx\) |
\(0.9105464642 - 0.04848815178i\) |
\(L(1)\) |
\(\approx\) |
\(0.9105464642 - 0.04848815178i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.161 + 0.986i)T \) |
| 3 | \( 1 + (0.267 - 0.963i)T \) |
| 5 | \( 1 + (-0.561 - 0.827i)T \) |
| 7 | \( 1 + (0.976 - 0.214i)T \) |
| 11 | \( 1 + (0.796 - 0.605i)T \) |
| 13 | \( 1 + (-0.856 + 0.515i)T \) |
| 17 | \( 1 + (0.976 + 0.214i)T \) |
| 19 | \( 1 + (-0.994 + 0.108i)T \) |
| 23 | \( 1 + (0.647 + 0.762i)T \) |
| 29 | \( 1 + (-0.161 - 0.986i)T \) |
| 31 | \( 1 + (-0.994 - 0.108i)T \) |
| 37 | \( 1 + (0.468 + 0.883i)T \) |
| 41 | \( 1 + (0.647 - 0.762i)T \) |
| 43 | \( 1 + (0.796 + 0.605i)T \) |
| 47 | \( 1 + (-0.561 + 0.827i)T \) |
| 53 | \( 1 + (0.907 + 0.419i)T \) |
| 61 | \( 1 + (-0.161 + 0.986i)T \) |
| 67 | \( 1 + (0.468 - 0.883i)T \) |
| 71 | \( 1 + (-0.561 + 0.827i)T \) |
| 73 | \( 1 + (0.0541 + 0.998i)T \) |
| 79 | \( 1 + (0.267 + 0.963i)T \) |
| 83 | \( 1 + (-0.725 - 0.687i)T \) |
| 89 | \( 1 + (-0.161 - 0.986i)T \) |
| 97 | \( 1 + (0.0541 - 0.998i)T \) |
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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
−32.556959282338099621442573924422, −31.433610782225745378616898235, −30.63633977933741748038715468662, −29.65948654057614760263644936542, −27.82967646970651119573473258624, −27.53128137582715314093108642223, −26.57315610665893355754309458573, −25.29420135508598035576402889067, −23.26515197777115316303325481667, −22.2869171514293496400762555316, −21.408284375898842809518505343163, −20.26494075853000325772840714877, −19.347146549106594584978827308836, −18.001927525898670879118018179615, −16.79457041951002497316265317832, −14.66213977116039953193352952527, −14.6368693622850167097415457076, −12.34097471599838367405789455263, −11.179596657456333621091706876648, −10.32033132782204160633376664667, −8.9849146178034643159555162788, −7.66597626972121917271504633036, −5.011936696130559543866111458941, −3.82875335344363583887463059559, −2.4255588626000041062379474363,
1.25825758017440830168087414108, 4.16438009712193320289016708134, 5.693549218175168299084904597150, 7.29515828888280951698953612417, 8.16384987885669460075132687874, 9.19332522206094118193750726999, 11.54591315108665122580163712661, 12.79584080619215702716363733249, 14.11460147587998532926253175222, 14.95647192633513116319843635937, 16.78581992844331907705895169127, 17.299794323974057227143691293605, 18.87919477043346747835759983656, 19.63353585004063153976672699182, 21.259130251081512381937260567580, 23.08329476930566949676256524835, 24.04609989679728088233072430193, 24.50094737112464591876477383353, 25.622119303367069739227338680588, 27.10553797446871264538003423603, 27.810459357345034338932983665366, 29.37009933931591986339666316253, 30.71908713580891784307648003823, 31.6594104770802139185698438063, 32.512505823543241356803786904762