L(s) = 1 | + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.342 − 0.939i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (0.342 − 0.939i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.342 − 0.939i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (0.342 − 0.939i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.182722695 + 0.0002011043476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182722695 + 0.0002011043476i\) |
\(L(1)\) |
\(\approx\) |
\(0.8840049476 + 0.06113331014i\) |
\(L(1)\) |
\(\approx\) |
\(0.8840049476 + 0.06113331014i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.984 + 0.173i)T \) |
| 7 | \( 1 + (0.642 + 0.766i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.342 - 0.939i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.642 - 0.766i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 - 0.939i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−20.90918827615569943451259578387, −19.77797313380560808529749029661, −19.078935317170616759220558282677, −18.330390964241645229157270845547, −17.51155725350095226508612101470, −17.04677268134415202138661736338, −16.25772771695732371424652394899, −15.590585917662249264440564904017, −14.59374356467564491195332193031, −13.487288044948858061944519359459, −13.31991217415406463939863329051, −12.110070459524518643981648402115, −11.297988027743597747785783712, −10.899717709193028021737655302711, −10.18944171709496635132196004901, −8.99933845145852832170079349822, −8.1827035963217714002948912669, −7.204583728816781529482734026128, −6.59508238679862803981602407581, −5.70309496594770855857570994083, −4.780547313332308885785778162951, −4.20762681410794234187567136589, −2.99494640251741933081729314685, −1.61761842587474815100814693418, −0.89150817391334026054160756108,
0.70484124576317025421230804168, 1.902534330532409222236167696, 2.87820891829326899133958923810, 4.17717810867726920092620513713, 5.03773525380345252385537436705, 5.48509872617453960755726927729, 6.44242396036660183872968261128, 7.35534662941207728891790000221, 8.177142955030186618379612619529, 9.12765069978747073681469706869, 10.17704356063714114030565119902, 10.56136828821057900055083303737, 11.70418127620655261872745626278, 12.04405438292760924367372734467, 12.885141131393946081006657435755, 13.73762726153515221878302631984, 14.95408700025705659527349984322, 15.417522099431616013481791518597, 16.03604734864922641748020969007, 17.10885760893651397729516962267, 17.62317499670561369694924820575, 18.42333865892512896023852708629, 18.71571799062782358600345022718, 20.13216193885315399488761749884, 20.89652791519070411215686565214