L(s) = 1 | + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.540 − 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯ |
L(s) = 1 | + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.540 − 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2369436397 + 0.2025462503i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2369436397 + 0.2025462503i\) |
\(L(1)\) |
\(\approx\) |
\(0.8000042473 - 0.2635459486i\) |
\(L(1)\) |
\(\approx\) |
\(0.8000042473 - 0.2635459486i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.755 - 0.654i)T \) |
| 11 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (0.540 - 0.841i)T \) |
| 17 | \( 1 + (0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.909 + 0.415i)T \) |
| 23 | \( 1 + (-0.909 + 0.415i)T \) |
| 29 | \( 1 + (0.755 + 0.654i)T \) |
| 31 | \( 1 + (-0.909 - 0.415i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.540 - 0.841i)T \) |
| 43 | \( 1 + (-0.755 + 0.654i)T \) |
| 47 | \( 1 + (0.841 - 0.540i)T \) |
| 53 | \( 1 + (0.841 + 0.540i)T \) |
| 59 | \( 1 + (0.540 + 0.841i)T \) |
| 61 | \( 1 + (-0.989 - 0.142i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.654 - 0.755i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.415 - 0.909i)T \) |
| 83 | \( 1 + (-0.281 + 0.959i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−22.032090471226925142268300400415, −21.32491761160856541839373400814, −20.5930047211980604744055348137, −19.90211394511382007233588326076, −19.1014949967585585199816894311, −18.3908636134181853252430297220, −16.93325183469925260095324417358, −16.33154182568063366209236670047, −15.62977669090190750467920380777, −15.10955887440562950009009496809, −14.01625354981155688643890104768, −13.05474793864072968495578343352, −12.26543316129918493050101405863, −11.42927174251713449972360284614, −10.2123861695964786811310140482, −9.58848986114853121536974019976, −8.64962004758836499215513752531, −8.13438636098916862618462212676, −6.895393337783384179002417669125, −5.63693113530828175855775548226, −4.717651655751875772565500425855, −3.962889855900719753227287454, −2.96212732113087381516225990963, −1.88687108295793544208623690477, −0.07966722340038188773094430705,
0.88654961803930610725726549158, 2.37070200875081035188445586657, 3.43400078994124849550452060115, 3.74094458016680562450504486637, 5.71161201053306428097724336985, 6.40079695988683131711032859293, 7.45117412066763139561237586866, 7.93405466593025841884783028027, 8.83664993807796428387064249059, 10.26372487256444377848854698395, 10.69736856021054540596592723794, 11.93957857932425347495633190944, 12.716177656284267433690301315829, 13.50342390358914632351691032417, 14.23608547592552397831464573847, 15.09759637941273045968775876260, 15.96029716874215357020256181027, 16.82059615546953588050120835362, 18.061463176991906305659291298823, 18.55649656554509505944202773365, 19.385466884326475413954589680752, 19.86286883578354327395763036638, 20.78256866191515216569400127373, 21.805284351796497229429193141845, 22.89495174642210920763123276041