L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.522 + 0.852i)11-s + (0.453 + 0.891i)12-s + (−0.996 + 0.0784i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.382 + 0.923i)17-s + (0.453 + 0.891i)18-s + (−0.760 − 0.649i)19-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.522 + 0.852i)11-s + (0.453 + 0.891i)12-s + (−0.996 + 0.0784i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.382 + 0.923i)17-s + (0.453 + 0.891i)18-s + (−0.760 − 0.649i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02053548614 + 0.02837928270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02053548614 + 0.02837928270i\) |
\(L(1)\) |
\(\approx\) |
\(0.5166716071 - 0.3340357719i\) |
\(L(1)\) |
\(\approx\) |
\(0.5166716071 - 0.3340357719i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 3 | \( 1 + (-0.156 - 0.987i)T \) |
| 7 | \( 1 + (-0.996 - 0.0784i)T \) |
| 11 | \( 1 + (0.522 + 0.852i)T \) |
| 13 | \( 1 + (-0.996 + 0.0784i)T \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
| 19 | \( 1 + (-0.760 - 0.649i)T \) |
| 23 | \( 1 + (-0.382 + 0.923i)T \) |
| 29 | \( 1 + (0.707 + 0.707i)T \) |
| 31 | \( 1 + (-0.923 + 0.382i)T \) |
| 37 | \( 1 + (0.382 - 0.923i)T \) |
| 41 | \( 1 + (0.987 - 0.156i)T \) |
| 43 | \( 1 + (0.0784 + 0.996i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.707 - 0.707i)T \) |
| 61 | \( 1 + (-0.707 - 0.707i)T \) |
| 67 | \( 1 + (-0.707 - 0.707i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (-0.996 - 0.0784i)T \) |
| 79 | \( 1 + (0.453 + 0.891i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.972 - 0.233i)T \) |
| 97 | \( 1 + T \) |
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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
−17.060286263828711463404028416327, −16.75539114129316973305042851045, −16.247820775613013446903960258799, −15.77002866772242703509892966541, −14.80310802514768518758683586946, −14.61083955840303081168311438188, −13.797271314257086768685239485450, −13.105980088881327393251346631468, −12.20169305793628308535165791307, −11.648952997092736415896478477930, −10.488686672413141156370846918398, −10.07153260944647756638246644873, −9.51590761700416461953928325807, −8.82986505755549416444209641756, −8.30793124027038856156710376405, −7.34489989033064781495561470364, −6.58769967736827897363746810298, −5.95238742648469713333911621653, −5.47728733598309594843548191660, −4.53994808421521284474475663448, −4.007253414107978978186208448204, −3.22816003077300542146347544274, −2.42491884293430987345050638730, −0.7574185464810386086821548638, −0.01411096724898980971575109529,
1.10832931116395780673830128777, 1.864816596132755565091185680548, 2.50670055293513776858097277232, 3.25407697395095827040874457566, 4.06502280895722091092378172086, 4.86453677503233803149557403185, 5.75495557220732548421331923933, 6.49298279104449576961016874690, 7.28360695653563823414233728167, 7.766339815048369663790526662173, 8.75546031210374822805833617990, 9.351435896111252783530712788221, 9.92586500598761504177987165301, 10.73712331709670976153533965157, 11.35052679493729481632491234978, 12.23956455214277999187093628939, 12.69360501357632858556736980525, 12.803563752502005814510957816173, 13.82794235029377583070776347454, 14.41675511918936529796668745166, 15.03431084675430377969271343585, 16.20471496160095048155640731141, 16.92841524365543685230906445005, 17.4401667075475137844832498252, 17.92065650423054764475793750322