L(s) = 1 | + (−0.273 + 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 − 0.526i)4-s + (−0.982 − 0.183i)5-s + (0.932 − 0.361i)6-s + (0.739 + 0.673i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (−0.273 + 0.961i)11-s + (0.0922 + 0.995i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (0.445 + 0.895i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + ⋯ |
L(s) = 1 | + (−0.273 + 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 − 0.526i)4-s + (−0.982 − 0.183i)5-s + (0.932 − 0.361i)6-s + (0.739 + 0.673i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (−0.273 + 0.961i)11-s + (0.0922 + 0.995i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (0.445 + 0.895i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0945 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0945 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3680853605 + 0.4047061827i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3680853605 + 0.4047061827i\) |
\(L(1)\) |
\(\approx\) |
\(0.5803520530 + 0.2634950716i\) |
\(L(1)\) |
\(\approx\) |
\(0.5803520530 + 0.2634950716i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (-0.273 + 0.961i)T \) |
| 3 | \( 1 + (-0.602 - 0.798i)T \) |
| 5 | \( 1 + (-0.982 - 0.183i)T \) |
| 7 | \( 1 + (0.739 + 0.673i)T \) |
| 11 | \( 1 + (-0.273 + 0.961i)T \) |
| 13 | \( 1 + (0.739 + 0.673i)T \) |
| 17 | \( 1 + (0.932 + 0.361i)T \) |
| 19 | \( 1 + (-0.602 + 0.798i)T \) |
| 23 | \( 1 + (-0.273 - 0.961i)T \) |
| 29 | \( 1 + (-0.982 - 0.183i)T \) |
| 31 | \( 1 + (0.445 + 0.895i)T \) |
| 37 | \( 1 + (0.0922 + 0.995i)T \) |
| 41 | \( 1 + (-0.982 + 0.183i)T \) |
| 43 | \( 1 + (0.0922 - 0.995i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.602 + 0.798i)T \) |
| 59 | \( 1 + (0.739 - 0.673i)T \) |
| 61 | \( 1 + (0.932 + 0.361i)T \) |
| 67 | \( 1 + (0.739 - 0.673i)T \) |
| 71 | \( 1 + (-0.982 + 0.183i)T \) |
| 73 | \( 1 + (-0.982 + 0.183i)T \) |
| 79 | \( 1 + (-0.982 - 0.183i)T \) |
| 83 | \( 1 + (0.739 + 0.673i)T \) |
| 89 | \( 1 + (-0.850 + 0.526i)T \) |
| 97 | \( 1 + (0.932 - 0.361i)T \) |
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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
−29.68951112768521612645258700537, −28.14499925213479408145209998072, −27.63605268610028964243106315774, −26.8717682429577375104432639806, −26.02340154599798791584093482224, −23.78507825309527445452005548161, −23.20475520987868955582293023998, −22.10723748536868670685258416084, −21.02909703242086354366244080843, −20.333931707625679876345909689716, −19.10443253936322796367840733207, −17.98558790265640432740333019747, −16.92357246004157153775074158777, −15.86970369743994690244922064125, −14.53957575416807716361066506762, −13.13950628786335529269623028916, −11.57455602266361302445892078356, −11.15546465350001730427348694657, −10.21834562549125920550892107832, −8.69703986971049732990259739641, −7.628077823567219585299176065042, −5.44006625945478897717697289959, −4.13140272591436135023312828453, −3.28459689224164605660212581605, −0.729797320171090458090053493412,
1.57406837601252276027012563662, 4.33238651802080697586618220105, 5.48438021180972694715307021182, 6.75567891048689087875411320660, 7.887227140110030224595771011519, 8.58022947910017724121685748855, 10.47490328561496163964664819403, 11.86549676702328262260823622820, 12.7221091221317447608607807425, 14.26161085037561859558625060743, 15.262027787491668103214941930861, 16.394305338233348360705266245129, 17.31889155926958373934091799374, 18.6405517222003402067262783810, 18.84609449674328882850259338092, 20.5566262367511788379199604543, 22.23334722521314797523968460031, 23.407128498179876799283912306734, 23.69739575276430011634164036550, 24.84547628345352043493148677480, 25.68625817663424939950865845019, 27.12563233379177789094465756072, 28.11357717121096830793305629296, 28.46241283658473347400031301464, 30.438732262160413331050383444458