L(s) = 1 | + (−0.354 + 0.935i)2-s + (0.120 + 0.992i)3-s + (−0.748 − 0.663i)4-s + (−0.970 + 0.239i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (0.885 − 0.464i)8-s + (−0.970 + 0.239i)9-s + (0.120 − 0.992i)10-s + (0.568 − 0.822i)11-s + (0.568 − 0.822i)12-s + (−0.748 + 0.663i)13-s + (−0.748 − 0.663i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (0.885 + 0.464i)17-s + ⋯ |
L(s) = 1 | + (−0.354 + 0.935i)2-s + (0.120 + 0.992i)3-s + (−0.748 − 0.663i)4-s + (−0.970 + 0.239i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (0.885 − 0.464i)8-s + (−0.970 + 0.239i)9-s + (0.120 − 0.992i)10-s + (0.568 − 0.822i)11-s + (0.568 − 0.822i)12-s + (−0.748 + 0.663i)13-s + (−0.748 − 0.663i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (0.885 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04793650305 + 0.5341174634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04793650305 + 0.5341174634i\) |
\(L(1)\) |
\(\approx\) |
\(0.4103233777 + 0.5356937286i\) |
\(L(1)\) |
\(\approx\) |
\(0.4103233777 + 0.5356937286i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (-0.354 + 0.935i)T \) |
| 3 | \( 1 + (0.120 + 0.992i)T \) |
| 5 | \( 1 + (-0.970 + 0.239i)T \) |
| 7 | \( 1 + (-0.354 + 0.935i)T \) |
| 11 | \( 1 + (0.568 - 0.822i)T \) |
| 13 | \( 1 + (-0.748 + 0.663i)T \) |
| 17 | \( 1 + (0.885 + 0.464i)T \) |
| 19 | \( 1 + (-0.748 + 0.663i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.568 + 0.822i)T \) |
| 31 | \( 1 + (0.568 + 0.822i)T \) |
| 37 | \( 1 + (0.120 + 0.992i)T \) |
| 41 | \( 1 + (0.568 - 0.822i)T \) |
| 43 | \( 1 + (0.120 - 0.992i)T \) |
| 47 | \( 1 + (-0.970 - 0.239i)T \) |
| 59 | \( 1 + (-0.970 - 0.239i)T \) |
| 61 | \( 1 + (0.885 - 0.464i)T \) |
| 67 | \( 1 + (-0.748 - 0.663i)T \) |
| 71 | \( 1 + (0.120 - 0.992i)T \) |
| 73 | \( 1 + (0.885 + 0.464i)T \) |
| 79 | \( 1 + (-0.354 - 0.935i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.885 + 0.464i)T \) |
| 97 | \( 1 + (-0.970 + 0.239i)T \) |
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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
−32.356730873732396335413652152679, −31.37971125856289786100772826696, −30.23627499831025673172693543506, −29.73009789009736815081312595247, −28.36246937753707088193080983767, −27.3404094712552959907391160531, −26.17473106296294412902071610843, −24.84380132004471656173943280206, −23.20394610716118066878050561537, −22.81154123281715177783120265516, −20.71647959172912549616448966043, −19.67835827860106556417698099412, −19.27467778162323373946340789163, −17.68384557900344202405591162766, −16.787911937272009441520070129499, −14.66865040062148031163433508733, −13.13696109623428329461697931472, −12.3298160148038102551313103368, −11.20537739655518055665214004279, −9.601375862849303974736530033211, −8.006487739365523644353784272302, −7.135576567471251730157398644961, −4.45980338516380275499274853275, −2.8945212949372291124191755653, −0.84217888060740760289442821386,
3.47380813115694702484227011387, 4.97389922988203504101127431329, 6.46073193479141728574752200087, 8.24566400431848730295491620586, 9.138998607071035207274173872, 10.56512632198935435676054538167, 12.10438758185552558109404251940, 14.33287076781283010941614658914, 15.072232199124506923612445427277, 16.14598236592983130356414115307, 16.949682075614479574116117315679, 18.91426243579941739592285886905, 19.462070745866109053845198447119, 21.474195493604732611465176268457, 22.46459424052235913053021607127, 23.571693861434072392880791144718, 24.949835046407358178993997817553, 26.02191523879944375393910764985, 27.16742247658275048375441595071, 27.62317598587406516313856462294, 28.93318619060869433623807779619, 31.1268809826543289560989372612, 31.88418971816748550817534538992, 32.6742984733135576625898578875, 34.27822130153401011454487176945