L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.258 − 0.965i)7-s − i·8-s + (0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.5 − 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + i·19-s + (0.965 + 0.258i)20-s + (0.258 + 0.965i)22-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s − i·26-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.258 − 0.965i)7-s − i·8-s + (0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.5 − 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + i·19-s + (0.965 + 0.258i)20-s + (0.258 + 0.965i)22-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s − i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.727766795 - 0.7042467891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.727766795 - 0.7042467891i\) |
\(L(1)\) |
\(\approx\) |
\(1.627486178 - 0.4678887172i\) |
\(L(1)\) |
\(\approx\) |
\(1.627486178 - 0.4678887172i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.258 + 0.965i)T \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
| 11 | \( 1 + (-0.258 + 0.965i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.965 - 0.258i)T \) |
| 29 | \( 1 + (-0.965 - 0.258i)T \) |
| 31 | \( 1 + (-0.258 - 0.965i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.965 + 0.258i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.258 - 0.965i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.258 + 0.965i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.965 - 0.258i)T \) |
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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
−28.378405654419700257011427460982, −27.03327652541167790043963555386, −25.84004492271639825926810104352, −24.98297645185382124225286732172, −24.15690402765880246474699382219, −23.56299336703732839787269000124, −22.03836101753429390041530614682, −21.40132070433677678042444301835, −20.688367454747963473986933226991, −19.29399009406416257397436248941, −17.97606080760412829390323244158, −16.80286658146455475208892946184, −16.0453710329869497221898046966, −15.10734743575244092294746976226, −13.82160756956026662656252748764, −13.08650471521553448512838796284, −11.99135974988506193411406972378, −11.117639206655412575699150912374, −8.980079975431535405852985372587, −8.515794145192471896995243645185, −6.9005325443491644942686590745, −5.60838567016903560940406857090, −4.96898108788390630230084271383, −3.46841208677565781768285222504, −1.94844898382228068528322992609,
1.62798305888753376657100596965, 3.043684960080100131522382451022, 4.11875406182920444238950352164, 5.48163066321243348521787622002, 6.71512837334890440706616797738, 7.68620654466162933909777027315, 9.86091476741630087765077018223, 10.51848482954489344893153412115, 11.41453843111104815049309413317, 12.81922013473652674030014795043, 13.64233856143213575143627410812, 14.70821657238014088706786531005, 15.33656640494776752961074377177, 16.907517752980387681405237597283, 18.140348572882738301331812095352, 19.05188096257783965666849188729, 20.42218623344554153892060561828, 20.7829571911741454960291252658, 22.227661604052912792545413648187, 22.90797071151450604725625544185, 23.54513900262730254044667339369, 24.9222558629244465530731349591, 25.80752228898530342624256036943, 27.00309566355073904309627610322, 28.007425796385457871809345222263