L(s) = 1 | + (0.971 − 0.235i)2-s + (−0.304 − 0.952i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.520 − 0.853i)6-s + (−0.560 + 0.828i)7-s + (0.755 − 0.654i)8-s + (−0.814 + 0.580i)9-s + (0.814 + 0.580i)10-s + (0.189 − 0.981i)11-s + (−0.707 − 0.707i)12-s + (−0.349 + 0.936i)14-s + (0.520 − 0.853i)15-s + (0.580 − 0.814i)16-s + (0.971 + 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯ |
L(s) = 1 | + (0.971 − 0.235i)2-s + (−0.304 − 0.952i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.520 − 0.853i)6-s + (−0.560 + 0.828i)7-s + (0.755 − 0.654i)8-s + (−0.814 + 0.580i)9-s + (0.814 + 0.580i)10-s + (0.189 − 0.981i)11-s + (−0.707 − 0.707i)12-s + (−0.349 + 0.936i)14-s + (0.520 − 0.853i)15-s + (0.580 − 0.814i)16-s + (0.971 + 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.770427477 - 0.8999329452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.770427477 - 0.8999329452i\) |
\(L(1)\) |
\(\approx\) |
\(1.846765606 - 0.4919184631i\) |
\(L(1)\) |
\(\approx\) |
\(1.846765606 - 0.4919184631i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.971 - 0.235i)T \) |
| 3 | \( 1 + (-0.304 - 0.952i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.560 + 0.828i)T \) |
| 11 | \( 1 + (0.189 - 0.981i)T \) |
| 17 | \( 1 + (0.971 + 0.235i)T \) |
| 19 | \( 1 + (-0.165 + 0.986i)T \) |
| 23 | \( 1 + (0.771 + 0.636i)T \) |
| 29 | \( 1 + (-0.436 - 0.899i)T \) |
| 31 | \( 1 + (0.349 - 0.936i)T \) |
| 37 | \( 1 + (-0.258 + 0.965i)T \) |
| 41 | \( 1 + (0.672 + 0.739i)T \) |
| 43 | \( 1 + (0.436 - 0.899i)T \) |
| 47 | \( 1 + (0.841 + 0.540i)T \) |
| 53 | \( 1 + (-0.540 - 0.841i)T \) |
| 59 | \( 1 + (0.952 + 0.304i)T \) |
| 61 | \( 1 + (-0.393 - 0.919i)T \) |
| 67 | \( 1 + (-0.458 + 0.888i)T \) |
| 71 | \( 1 + (0.327 + 0.945i)T \) |
| 73 | \( 1 + (0.909 - 0.415i)T \) |
| 79 | \( 1 + (-0.909 + 0.415i)T \) |
| 83 | \( 1 + (0.479 - 0.877i)T \) |
| 97 | \( 1 + (-0.189 - 0.981i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.31793033085642770324840577095, −20.80042354285205740200806568942, −20.16783962224577212928885631550, −19.534439301023848343687735753451, −17.78648771375632873361555000062, −17.2004510165878889632882614682, −16.52802930930364873199594516761, −16.04716452005670374752470210711, −15.1220466730132278633453503817, −14.32292699234707299976909576738, −13.68749336710978139934669773380, −12.57450990868089261973925876960, −12.35969780022980332873957815656, −11.013824177680186759956635946433, −10.42135257329450943444442122753, −9.535234032314373039704639768327, −8.7981059669817887909421837418, −7.41758677365609992165058223939, −6.67268196160270805486420413428, −5.72283239252705439180418242798, −4.922354074364011514194383537909, −4.400915992540874321590084854815, −3.467327666359779960395495706528, −2.4951073230262495406757642564, −1.05883840881032250393270445950,
1.15953656433752381831124643884, 2.16488983964156382672538663783, 2.91820372557143215700178122009, 3.65089733889334965268070628746, 5.328352724375286748844439908026, 5.9391372796464984198925154118, 6.25889054138157715661463248026, 7.268892633269095283259621136782, 8.17665485041250155326737459050, 9.49314310220654464164188421263, 10.34040817511005288527764783833, 11.32215865309073343192811093920, 11.798417845715031603209873746964, 12.745311695976155977352858666, 13.31112588852141335269161292481, 14.07172661110081947739542597341, 14.68988786282740685483167308113, 15.58597581670310979530510086135, 16.64127923223851994263789060837, 17.2306227699414857065707442702, 18.53966075383693479659070512038, 18.994466268504664990468377442843, 19.28519817478912424404013761520, 20.69617823783155247658752667047, 21.361469483705218314308231548911