L(s) = 1 | + (0.913 − 0.406i)2-s + (−0.809 − 0.587i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.978 − 0.207i)6-s + (−0.104 − 0.994i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.978 + 0.207i)10-s + 11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (0.669 − 0.743i)17-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)2-s + (−0.809 − 0.587i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.978 − 0.207i)6-s + (−0.104 − 0.994i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.978 + 0.207i)10-s + 11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (0.669 − 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6875490848 - 0.7897652997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6875490848 - 0.7897652997i\) |
\(L(1)\) |
\(\approx\) |
\(0.9892627041 - 0.6261686610i\) |
\(L(1)\) |
\(\approx\) |
\(0.9892627041 - 0.6261686610i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (-0.104 - 0.994i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.104 + 0.994i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (0.669 + 0.743i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.978 - 0.207i)T \) |
| 71 | \( 1 + (-0.978 + 0.207i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (0.913 - 0.406i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.913 + 0.406i)T \) |
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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
−32.58260743243077099699913142599, −32.07421996419546724926070716062, −30.73966102256740436737904657764, −29.7827374505201989722466005499, −28.275739688608675423801993745676, −27.40632562023941544335971584213, −26.111869534941898364796627158620, −24.69513672789121056469349256410, −23.707085084411066541210454946685, −22.43066478807394558078225302114, −22.15804922431664495386369001068, −20.71896613447820047151831630204, −19.29842097636756757468120424664, −17.55450516782462569901528267481, −16.41382022716351762281005077714, −15.309270001931630393598254071793, −14.766038768376910848606759981343, −12.52778499952691095461353091121, −11.93183985879449815654354218250, −10.73330544657357699962430597124, −8.71117246874129441081946646371, −7.01485583856264559681231868273, −5.75452662891243212839512677263, −4.49486700137782767066880993773, −3.17656661990842307204122755946,
1.30499332562951839294003782689, 3.72105063541892320320723026821, 4.8714354411011693249269924218, 6.58048482884531397398314072526, 7.5249337255794028045815095298, 10.0235653595625192210803470392, 11.552860107687571875430485803555, 11.95143660642825875270572937368, 13.366924611868131323995351131045, 14.49881916386249512336111149213, 16.168957364373397158010942682650, 16.985031351219552936520478120239, 18.9973970469045437658195812336, 19.61720991673779886953209666514, 20.96617808642171512589348096296, 22.51684153131055543492135977451, 23.14545907258432597073317228226, 24.00169316907028842203689475415, 24.970892078992912626656501708153, 27.061957568123319224377581430151, 28.00396350667472446370988501171, 29.306672232519587603492646757115, 29.958635186781433818833120359068, 31.011482245131259063748344404519, 32.08501015438128086671284825593