L(s) = 1 | + (0.935 + 0.353i)2-s + (−0.999 + 0.0361i)3-s + (0.750 + 0.661i)4-s + (0.958 + 0.284i)5-s + (−0.947 − 0.319i)6-s + (0.590 − 0.806i)7-s + (0.468 + 0.883i)8-s + (0.997 − 0.0721i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.773 − 0.633i)12-s + (−0.891 − 0.452i)13-s + (0.837 − 0.546i)14-s + (−0.968 − 0.250i)15-s + (0.126 + 0.992i)16-s + (0.468 − 0.883i)17-s + ⋯ |
L(s) = 1 | + (0.935 + 0.353i)2-s + (−0.999 + 0.0361i)3-s + (0.750 + 0.661i)4-s + (0.958 + 0.284i)5-s + (−0.947 − 0.319i)6-s + (0.590 − 0.806i)7-s + (0.468 + 0.883i)8-s + (0.997 − 0.0721i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.773 − 0.633i)12-s + (−0.891 − 0.452i)13-s + (0.837 − 0.546i)14-s + (−0.968 − 0.250i)15-s + (0.126 + 0.992i)16-s + (0.468 − 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.975927380 + 0.7444103245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.975927380 + 0.7444103245i\) |
\(L(1)\) |
\(\approx\) |
\(1.618291116 + 0.4208572669i\) |
\(L(1)\) |
\(\approx\) |
\(1.618291116 + 0.4208572669i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 \) |
good | 2 | \( 1 + (0.935 + 0.353i)T \) |
| 3 | \( 1 + (-0.999 + 0.0361i)T \) |
| 5 | \( 1 + (0.958 + 0.284i)T \) |
| 7 | \( 1 + (0.590 - 0.806i)T \) |
| 11 | \( 1 + (0.267 + 0.963i)T \) |
| 13 | \( 1 + (-0.891 - 0.452i)T \) |
| 17 | \( 1 + (0.468 - 0.883i)T \) |
| 19 | \( 1 + (0.336 - 0.941i)T \) |
| 23 | \( 1 + (-0.891 - 0.452i)T \) |
| 29 | \( 1 + (-0.773 + 0.633i)T \) |
| 31 | \( 1 + (0.976 - 0.214i)T \) |
| 37 | \( 1 + (0.0541 + 0.998i)T \) |
| 41 | \( 1 + (0.468 + 0.883i)T \) |
| 43 | \( 1 + (0.700 - 0.713i)T \) |
| 47 | \( 1 + (-0.370 + 0.928i)T \) |
| 53 | \( 1 + (-0.856 + 0.515i)T \) |
| 59 | \( 1 + (-0.999 + 0.0361i)T \) |
| 61 | \( 1 + (-0.161 + 0.986i)T \) |
| 67 | \( 1 + (0.647 - 0.762i)T \) |
| 71 | \( 1 + (-0.999 - 0.0361i)T \) |
| 73 | \( 1 + (0.403 + 0.915i)T \) |
| 79 | \( 1 + (-0.856 - 0.515i)T \) |
| 83 | \( 1 + (-0.232 - 0.972i)T \) |
| 89 | \( 1 + (-0.0901 - 0.995i)T \) |
| 97 | \( 1 + (0.958 - 0.284i)T \) |
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
−24.44151948725457647028166019274, −24.00443107984745150621564144635, −22.77672858281375987086072075325, −21.92253737296938096622766187249, −21.476457659729703782350531567928, −20.880054500855743348595226020673, −19.354970565752533142883390078524, −18.60431971610491137622454000531, −17.49652023849197748384615266790, −16.67760623137797884820981259744, −15.80132156200586970785204806724, −14.57615574815463006779004650792, −13.90918115886586262882196953138, −12.71355750859246165347010952218, −12.094893911775709347209379409200, −11.29291777613898358922384899909, −10.26898167823672558147061821367, −9.42017489608840969233934235849, −7.81145400301388348472232491936, −6.28080024898363471077646765024, −5.78593137626546064141675507234, −5.06457718246371742996108605347, −3.88443889395475495657563505612, −2.23263337498643792284416305571, −1.382189189822857105722021249453,
1.49545165303549240898243296196, 2.793665764909179286846576232796, 4.492511525218399934974927117371, 4.956273425151800467865422564, 6.03606711208107763562039373333, 7.02993559023578708668459965641, 7.59635663429424726465713522308, 9.62771394537232798006501045370, 10.437790749212156451877748185253, 11.41593619109575608282951602614, 12.29704459240804491250420431211, 13.20890551243477406764417715964, 14.12586035392415561435705421708, 14.90524190592538069455353806015, 16.00238191014704300175514102911, 17.13425825309913509085988767090, 17.39099831762406919765291201643, 18.30730841993409034765757098213, 20.07968718546694377531055008448, 20.745555092371878693102100613, 21.73517688608064428372397447894, 22.45207184088623430649622878503, 22.95348972770435602659403452011, 24.10347646082447263565418670283, 24.57522739318766603239457195140