L(s) = 1 | + (0.934 − 0.355i)2-s + (0.969 + 0.245i)3-s + (0.746 − 0.665i)4-s + (−0.603 + 0.797i)5-s + (0.993 − 0.115i)6-s + (0.997 − 0.0709i)7-s + (0.461 − 0.887i)8-s + (0.879 + 0.476i)9-s + (−0.280 + 0.959i)10-s + (0.305 + 0.952i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (0.906 − 0.421i)14-s + (−0.781 + 0.624i)15-s + (0.115 − 0.993i)16-s + (−0.740 + 0.671i)17-s + ⋯ |
L(s) = 1 | + (0.934 − 0.355i)2-s + (0.969 + 0.245i)3-s + (0.746 − 0.665i)4-s + (−0.603 + 0.797i)5-s + (0.993 − 0.115i)6-s + (0.997 − 0.0709i)7-s + (0.461 − 0.887i)8-s + (0.879 + 0.476i)9-s + (−0.280 + 0.959i)10-s + (0.305 + 0.952i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (0.906 − 0.421i)14-s + (−0.781 + 0.624i)15-s + (0.115 − 0.993i)16-s + (−0.740 + 0.671i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.535751794 + 2.833533731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.535751794 + 2.833533731i\) |
\(L(1)\) |
\(\approx\) |
\(2.490154279 + 0.3983309515i\) |
\(L(1)\) |
\(\approx\) |
\(2.490154279 + 0.3983309515i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 709 | \( 1 \) |
good | 2 | \( 1 + (0.934 - 0.355i)T \) |
| 3 | \( 1 + (0.969 + 0.245i)T \) |
| 5 | \( 1 + (-0.603 + 0.797i)T \) |
| 7 | \( 1 + (0.997 - 0.0709i)T \) |
| 11 | \( 1 + (0.305 + 0.952i)T \) |
| 13 | \( 1 + (-0.899 + 0.437i)T \) |
| 17 | \( 1 + (-0.740 + 0.671i)T \) |
| 19 | \( 1 + (0.0443 + 0.999i)T \) |
| 23 | \( 1 + (-0.838 + 0.545i)T \) |
| 29 | \( 1 + (0.924 + 0.380i)T \) |
| 31 | \( 1 + (-0.988 + 0.150i)T \) |
| 37 | \( 1 + (0.0709 + 0.997i)T \) |
| 41 | \( 1 + (0.967 + 0.254i)T \) |
| 43 | \( 1 + (0.0620 - 0.998i)T \) |
| 47 | \( 1 + (0.998 - 0.0532i)T \) |
| 53 | \( 1 + (-0.364 + 0.931i)T \) |
| 59 | \( 1 + (-0.887 - 0.461i)T \) |
| 61 | \( 1 + (-0.995 + 0.0974i)T \) |
| 67 | \( 1 + (-0.468 - 0.883i)T \) |
| 71 | \( 1 + (-0.280 - 0.959i)T \) |
| 73 | \( 1 + (-0.106 - 0.994i)T \) |
| 79 | \( 1 + (0.651 - 0.758i)T \) |
| 83 | \( 1 + (0.786 + 0.617i)T \) |
| 89 | \( 1 + (0.228 + 0.973i)T \) |
| 97 | \( 1 + (0.567 - 0.823i)T \) |
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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
−22.10023076922255194539975819267, −21.45925809927067024363279637461, −20.64399563688713491339966480281, −19.94659430740776782037690338050, −19.49086666224893755619934634451, −18.086631929098316166809295337206, −17.27282826586155376289537859317, −16.14118963466946456328148767025, −15.61016277024107315686066151878, −14.67667537081268905417981684344, −14.09853817510487919947212881053, −13.27361262656929318063871817625, −12.466072530738771978262634113748, −11.70223118008682305061890706339, −10.86988445853123962966393927760, −9.21474192593526025991165272443, −8.49049052336909097282035627320, −7.76540377951950819928465940328, −7.09438268755628693930223298658, −5.75966904027124527630089846357, −4.66041264903179971753890449115, −4.18514007633732436538512662129, −2.95165782656660292937618740313, −2.104452199863818210156627689135, −0.7071789379264768454548913253,
1.6866932591155753951328090032, 2.222754283209786486158866943156, 3.40004009729960527280068347290, 4.25114385432396929740446371618, 4.76424917160278141344477630946, 6.28445679894988138757060301858, 7.37359919410377546206206168511, 7.791189580374892373354210098746, 9.176169574773636005923104138938, 10.25839984151205222412178646731, 10.78374435359907443490990977825, 11.9681112287085423737559171812, 12.429798332586427023471473870123, 13.884043156729136087749963389423, 14.23662581961172349928815067353, 15.11717441059056921975085028447, 15.30997408049523010999812765449, 16.56841430150207215182907655843, 17.88021775494725944099759005687, 18.78026425569590968738516036865, 19.70272312453144618637982922156, 20.06969219199509281142342386191, 20.933745193344310158931143482586, 21.88779449612160928397324636028, 22.20369734227073767834457800816