L(s) = 1 | + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.989 − 0.142i)12-s + (0.540 − 0.841i)13-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.755 + 0.654i)18-s + (0.959 − 0.281i)19-s + ⋯ |
L(s) = 1 | + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.989 − 0.142i)12-s + (0.540 − 0.841i)13-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.755 + 0.654i)18-s + (0.959 − 0.281i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.758855231 - 0.4381487513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.758855231 - 0.4381487513i\) |
\(L(1)\) |
\(\approx\) |
\(1.720125876 - 0.2257620226i\) |
\(L(1)\) |
\(\approx\) |
\(1.720125876 - 0.2257620226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.989 - 0.142i)T \) |
| 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (0.540 + 0.841i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.540 - 0.841i)T \) |
| 17 | \( 1 + (0.281 - 0.959i)T \) |
| 19 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.959 + 0.281i)T \) |
| 31 | \( 1 + (0.415 + 0.909i)T \) |
| 37 | \( 1 + (0.755 - 0.654i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.909 - 0.415i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.540 - 0.841i)T \) |
| 59 | \( 1 + (-0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (0.989 - 0.142i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (0.281 + 0.959i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.755 + 0.654i)T \) |
| 89 | \( 1 + (-0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.755 - 0.654i)T \) |
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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
−29.212874281456568270922162194470, −28.34341563720400407443807428879, −26.93905886738825923946076078473, −26.17264749241244581118975417890, −24.52514353723882726226697661447, −23.72962859090736410705157818046, −23.13250961572717695955263080787, −21.856028550240879699725670907705, −21.24898575495171549457266058037, −20.23978420066603520419374316933, −18.679591754304810316360680287337, −17.149910185724753154088460213874, −16.52188823079766527414491657179, −15.50672740066571137587039353092, −14.21368794228368475219915287164, −13.321476443927135410182940833129, −11.8658882314747238434662324234, −11.1662446742438440856550033013, −10.16053207465530470997589487636, −8.12034898946739774411787525792, −6.70241342381941409702131057453, −5.73030040773395435773952073590, −4.504465445549450310932510005452, −3.54214721561077202187967566829, −1.26589440132915264756451847978,
1.32300238008629431554583805252, 2.80050193627995093985566475148, 4.78531450237100148597263136256, 5.43679096084353175491841876747, 6.71219314862162225170296447995, 7.8644736257097179078602209457, 9.95853498077745230892774878054, 11.20479648338005013219565859528, 12.03807753719764429105744913955, 12.84089689954703301628168139997, 14.06466009928982789975316557993, 15.39097371670412448232951762232, 16.1271189627857053615828903807, 17.69644713963558546907100710874, 18.426992501697082159336505476112, 19.92911453481640320785720680723, 20.9995645212986856534890879861, 22.04708204175794694285499781898, 22.872016795826198860143684171064, 23.626853720400878595249765340548, 24.85974727605347986150432382879, 25.27442735703071926180034974274, 27.329162849357931697741396975337, 28.36738137864098005699382041898, 28.92291593680881884026772807670