L(s) = 1 | + (−0.813 + 0.580i)3-s + (0.734 − 0.678i)5-s + (0.112 − 0.993i)7-s + (0.324 − 0.945i)9-s + (0.957 − 0.289i)11-s + (−0.717 + 0.696i)13-s + (−0.203 + 0.979i)15-s + (−0.970 − 0.240i)17-s + (0.935 + 0.352i)19-s + (0.485 + 0.874i)21-s + (−0.998 − 0.0586i)23-s + (0.0795 − 0.996i)25-s + (0.285 + 0.958i)27-s + (−0.906 − 0.421i)29-s + (0.999 + 0.0167i)31-s + ⋯ |
L(s) = 1 | + (−0.813 + 0.580i)3-s + (0.734 − 0.678i)5-s + (0.112 − 0.993i)7-s + (0.324 − 0.945i)9-s + (0.957 − 0.289i)11-s + (−0.717 + 0.696i)13-s + (−0.203 + 0.979i)15-s + (−0.970 − 0.240i)17-s + (0.935 + 0.352i)19-s + (0.485 + 0.874i)21-s + (−0.998 − 0.0586i)23-s + (0.0795 − 0.996i)25-s + (0.285 + 0.958i)27-s + (−0.906 − 0.421i)29-s + (0.999 + 0.0167i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1002100667 - 0.5849862356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1002100667 - 0.5849862356i\) |
\(L(1)\) |
\(\approx\) |
\(0.8216560195 - 0.1317065752i\) |
\(L(1)\) |
\(\approx\) |
\(0.8216560195 - 0.1317065752i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (-0.813 + 0.580i)T \) |
| 5 | \( 1 + (0.734 - 0.678i)T \) |
| 7 | \( 1 + (0.112 - 0.993i)T \) |
| 11 | \( 1 + (0.957 - 0.289i)T \) |
| 13 | \( 1 + (-0.717 + 0.696i)T \) |
| 17 | \( 1 + (-0.970 - 0.240i)T \) |
| 19 | \( 1 + (0.935 + 0.352i)T \) |
| 23 | \( 1 + (-0.998 - 0.0586i)T \) |
| 29 | \( 1 + (-0.906 - 0.421i)T \) |
| 31 | \( 1 + (0.999 + 0.0167i)T \) |
| 37 | \( 1 + (-0.521 - 0.853i)T \) |
| 41 | \( 1 + (0.728 + 0.684i)T \) |
| 43 | \( 1 + (-0.0125 + 0.999i)T \) |
| 47 | \( 1 + (0.418 + 0.908i)T \) |
| 53 | \( 1 + (-0.876 - 0.481i)T \) |
| 59 | \( 1 + (-0.932 + 0.360i)T \) |
| 61 | \( 1 + (0.0209 + 0.999i)T \) |
| 67 | \( 1 + (0.121 - 0.992i)T \) |
| 71 | \( 1 + (0.212 - 0.977i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.935 + 0.352i)T \) |
| 83 | \( 1 + (-0.783 - 0.621i)T \) |
| 89 | \( 1 + (0.859 - 0.510i)T \) |
| 97 | \( 1 + (0.804 - 0.594i)T \) |
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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
−17.88905566371426681313744687760, −17.34619439152167012261915282940, −17.17048291653300582789214432294, −15.92879673125888395486464782801, −15.465731930282627691689335548337, −14.70637591086844378199212021545, −14.00632676129857426410014749844, −13.41811051210137709486335165131, −12.61358139464107910830990046033, −12.03808765415216419921084079364, −11.5358148496126836478241125669, −10.80587327233791024837854998804, −10.08182777342341126347318702361, −9.46641750116859346919697707173, −8.7220541386656245421586275860, −7.78017222180288294950026106240, −7.07107541620471448246853056620, −6.50179259289988816874357794818, −5.85200431092178077695812766542, −5.32063210237941737991665038498, −4.59177832695612062346791819811, −3.463584435493505514164893843891, −2.48016455622509924448378301029, −2.03488621156435753122242055781, −1.22764917296639144015621778804,
0.16862415309240570248861570384, 1.175691077250741302000850909041, 1.76087656379953286181985121721, 2.967246705081403772704012974095, 4.17948788209087794961540389716, 4.27283777191114620112467196870, 5.073598402264354896016536523394, 5.97865984584012413261819653884, 6.40084689487720647488839152330, 7.19509134228436962245864467877, 8.01087087257437977121856304257, 9.15988404093680127383155178779, 9.43716942572429119362671668621, 10.03191257833702988781883952899, 10.80132975203566707121049861339, 11.52977738984144788651833095509, 11.98116656591246266145746237171, 12.77939173288075321341336720728, 13.54961002599155421194662695704, 14.19325582108805304669601399887, 14.60388184461253653626471228673, 15.844570753674511433241349213888, 16.18630628178845319773011853122, 16.86037825905299097740002279013, 17.32309586529899192799178282676