L(s) = 1 | + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.327 − 0.945i)7-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)12-s + (−1.88 + 0.363i)13-s + (0.415 − 0.909i)16-s + (1.50 + 0.442i)19-s + (0.0475 − 0.998i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (−0.786 − 0.618i)28-s + (−0.544 + 0.627i)31-s + (0.981 + 0.189i)36-s − 0.654·37-s + ⋯ |
L(s) = 1 | + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.327 − 0.945i)7-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)12-s + (−1.88 + 0.363i)13-s + (0.415 − 0.909i)16-s + (1.50 + 0.442i)19-s + (0.0475 − 0.998i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (−0.786 − 0.618i)28-s + (−0.544 + 0.627i)31-s + (0.981 + 0.189i)36-s − 0.654·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.674777733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674777733\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.928 - 0.371i)T \) |
| 7 | \( 1 + (0.327 + 0.945i)T \) |
| 67 | \( 1 + (0.327 - 0.945i)T \) |
good | 2 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 11 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (1.88 - 0.363i)T + (0.928 - 0.371i)T^{2} \) |
| 17 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 19 | \( 1 + (-1.50 - 0.442i)T + (0.841 + 0.540i)T^{2} \) |
| 23 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + 0.654T + T^{2} \) |
| 41 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 43 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 53 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 59 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 61 | \( 1 + (0.653 + 1.43i)T + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 73 | \( 1 + (1.15 - 1.62i)T + (-0.327 - 0.945i)T^{2} \) |
| 79 | \( 1 + (0.642 - 0.123i)T + (0.928 - 0.371i)T^{2} \) |
| 83 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 89 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 97 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.965767547055557872681611970765, −9.128885935506702226511635979779, −7.943971312054390655419046785159, −7.11673532636789034541500952286, −6.95199229544241302733438700383, −5.38675228255060707376411559370, −4.68922681380966415017445012809, −3.45792034924537326809677821364, −2.68701209034255929600578114082, −1.52246922409557189754002494236,
1.83980521490681230409088261985, 2.86511618599689854562210566066, 3.15938560766989107371438151341, 4.71734281141685859263710553239, 5.74881683777848139250449080899, 6.88376969033395920596786031013, 7.35539547125313896453366203587, 8.045629328939717679487856717304, 8.993188355423891988056438216899, 9.591488507201809707903616747246