L(s) = 1 | + (0.947 + 1.04i)2-s + (−0.204 + 1.98i)4-s + (1.48 − 3.58i)5-s + (1.03 + 1.03i)7-s + (−2.28 + 1.67i)8-s + (5.17 − 1.84i)10-s + (2.98 + 1.23i)11-s + (1.33 + 3.23i)13-s + (−0.106 + 2.07i)14-s + (−3.91 − 0.811i)16-s − 5.31i·17-s + (−0.339 − 0.820i)19-s + (6.83 + 3.68i)20-s + (1.53 + 4.31i)22-s + (−4.32 + 4.32i)23-s + ⋯ |
L(s) = 1 | + (0.670 + 0.742i)2-s + (−0.102 + 0.994i)4-s + (0.664 − 1.60i)5-s + (0.392 + 0.392i)7-s + (−0.806 + 0.590i)8-s + (1.63 − 0.581i)10-s + (0.901 + 0.373i)11-s + (0.371 + 0.897i)13-s + (−0.0283 + 0.554i)14-s + (−0.979 − 0.202i)16-s − 1.28i·17-s + (−0.0779 − 0.188i)19-s + (1.52 + 0.825i)20-s + (0.326 + 0.919i)22-s + (−0.901 + 0.901i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86843 + 0.709932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86843 + 0.709932i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.947 - 1.04i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.48 + 3.58i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 1.03i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.98 - 1.23i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 3.23i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 5.31iT - 17T^{2} \) |
| 19 | \( 1 + (0.339 + 0.820i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (4.32 - 4.32i)T - 23iT^{2} \) |
| 29 | \( 1 + (5.78 - 2.39i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + (-0.646 + 1.56i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (3.42 - 3.42i)T - 41iT^{2} \) |
| 43 | \( 1 + (-6.50 - 2.69i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + (6.63 + 2.74i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.185 + 0.447i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (3.16 - 1.31i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (6.09 - 2.52i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (2.91 + 2.91i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1.02 + 1.02i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 + (-6.41 - 15.4i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.991 - 0.991i)T + 89iT^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−12.00757299583185371782205784771, −11.56073399647114845847777366952, −9.399258452902842072564810731347, −9.148689761851566960140775183650, −8.092261641630055995736983297138, −6.87366675469937205935923797528, −5.69917214197403493365451599657, −4.95411035880442409829902111213, −3.98399126522079132087808577867, −1.85475342973304400012276642335,
1.81665108340031635266274198378, 3.17065857586267944891158310294, 4.11837226721940848957893834194, 5.94626076051208387548690454674, 6.28138801614805471001391713246, 7.69032682559617646515228451226, 9.187504226725014339055281203743, 10.43843170340410243927990620618, 10.60908691620370857699390782713, 11.52243440773831856369784296841