L(s) = 1 | + (0.962 − 0.269i)2-s + (0.203 − 0.979i)3-s + (0.854 − 0.519i)4-s + (0.460 − 0.887i)5-s + (−0.0682 − 0.997i)6-s + (−0.576 + 0.816i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.334 − 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 + 0.631i)17-s + ⋯ |
L(s) = 1 | + (0.962 − 0.269i)2-s + (0.203 − 0.979i)3-s + (0.854 − 0.519i)4-s + (0.460 − 0.887i)5-s + (−0.0682 − 0.997i)6-s + (−0.576 + 0.816i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.334 − 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 + 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07237183881 - 2.110895410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07237183881 - 2.110895410i\) |
\(L(1)\) |
\(\approx\) |
\(1.211025080 - 1.145482689i\) |
\(L(1)\) |
\(\approx\) |
\(1.211025080 - 1.145482689i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.962 - 0.269i)T \) |
| 3 | \( 1 + (0.203 - 0.979i)T \) |
| 5 | \( 1 + (0.460 - 0.887i)T \) |
| 7 | \( 1 + (-0.576 + 0.816i)T \) |
| 11 | \( 1 + (-0.917 - 0.398i)T \) |
| 13 | \( 1 + (-0.917 + 0.398i)T \) |
| 17 | \( 1 + (-0.775 + 0.631i)T \) |
| 19 | \( 1 + (0.203 - 0.979i)T \) |
| 23 | \( 1 + (0.854 - 0.519i)T \) |
| 29 | \( 1 + (-0.917 + 0.398i)T \) |
| 31 | \( 1 + (0.682 - 0.730i)T \) |
| 37 | \( 1 + (-0.917 + 0.398i)T \) |
| 41 | \( 1 + (-0.0682 - 0.997i)T \) |
| 43 | \( 1 + (-0.775 - 0.631i)T \) |
| 47 | \( 1 + (-0.775 - 0.631i)T \) |
| 53 | \( 1 + (0.854 - 0.519i)T \) |
| 59 | \( 1 + (0.854 + 0.519i)T \) |
| 61 | \( 1 + (-0.0682 - 0.997i)T \) |
| 67 | \( 1 + (0.682 + 0.730i)T \) |
| 71 | \( 1 + (0.962 + 0.269i)T \) |
| 73 | \( 1 + (0.854 + 0.519i)T \) |
| 79 | \( 1 + (0.854 + 0.519i)T \) |
| 83 | \( 1 + (-0.0682 - 0.997i)T \) |
| 89 | \( 1 + (0.682 + 0.730i)T \) |
| 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
−22.31379812119014928130722765374, −21.35081242351899920753202828355, −20.86410037799697038777187083719, −20.015948770874612844985970372187, −19.33883766045523475167835702043, −17.96699146015684508160029428528, −17.15150681875498777379493605668, −16.434980739842361804967263121827, −15.5531432418150159071303461709, −15.02821758172633018867483416702, −14.23862432296183330029601514759, −13.55465251716775702014643576822, −12.85195521251187228703122979460, −11.61430310778592084953664150386, −10.768527644558594792417286420998, −10.1789338281544561363475213252, −9.489706484631818109112323587209, −7.94732658237737060371134587965, −7.262514416999327789210973036414, −6.3987502872752072094858657777, −5.33612536040696668165661561209, −4.729792825152828733379758962583, −3.5704564315922884361744875335, −3.00655850044661473352217911513, −2.15051528258202014377669890789,
0.551541223748226374891493105445, 2.097973487673026291711398261714, 2.36737280667299377634004845883, 3.51789207166036700130045487021, 4.99631239765483646318695872785, 5.40960456552082829464371502962, 6.442123968503665950279952870212, 7.0745406027188939908693390396, 8.34979754795176047059279556326, 9.06979028067158302076628944684, 10.075589081019038828837801510086, 11.27902743315849386075522959185, 12.03862905892931185473721751680, 12.83595595685791971096102271688, 13.147760494277530680125994721237, 13.85178575019942236672567051823, 14.96882142855924621136162915170, 15.574201957748981988415156954270, 16.62549311654975571115049436552, 17.35488829852021808013137687581, 18.50156438708526944681319073312, 19.164615028708623851478221241023, 19.84692856389972000347275543118, 20.60861324976795643995643084800, 21.43088957757398893544136027709