L(s) = 1 | + (44.8 + 5.79i)2-s + (353. − 228. i)3-s + (1.98e3 + 520. i)4-s + 1.26e4i·5-s + (1.71e4 − 8.22e3i)6-s − 3.21e4i·7-s + (8.58e4 + 3.48e4i)8-s + (7.24e4 − 1.61e5i)9-s + (−7.33e4 + 5.67e5i)10-s + 2.50e5·11-s + (8.18e5 − 2.69e5i)12-s − 9.76e5·13-s + (1.86e5 − 1.44e6i)14-s + (2.89e6 + 4.46e6i)15-s + (3.65e6 + 2.06e6i)16-s − 2.81e6i·17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.128i)2-s + (0.839 − 0.543i)3-s + (0.967 + 0.254i)4-s + 1.81i·5-s + (0.902 − 0.431i)6-s − 0.723i·7-s + (0.926 + 0.376i)8-s + (0.408 − 0.912i)9-s + (−0.232 + 1.79i)10-s + 0.469·11-s + (0.949 − 0.312i)12-s − 0.729·13-s + (0.0927 − 0.717i)14-s + (0.984 + 1.51i)15-s + (0.870 + 0.491i)16-s − 0.481i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(3.87953 + 0.621748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.87953 + 0.621748i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-44.8 - 5.79i)T \) |
| 3 | \( 1 + (-353. + 228. i)T \) |
good | 5 | \( 1 - 1.26e4iT - 4.88e7T^{2} \) |
| 7 | \( 1 + 3.21e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 2.50e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.76e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.81e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 6.17e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 3.69e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.70e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 2.18e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 4.00e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.38e7iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 9.95e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 4.43e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.45e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 - 5.47e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.88e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.87e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 2.17e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.38e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.31e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 - 1.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.51e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 3.62e10T + 7.15e21T^{2} \) |
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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
−17.63195023825653205579989904006, −15.47041089648220338864060316452, −14.33720618314680634074002211749, −13.77586770633099182659910568766, −11.89495422110388332244568319861, −10.30765920931882259265690848614, −7.46001726518005520430186019956, −6.64232775418364914285596803205, −3.68014673903788323729504833592, −2.37320684693563375402999678851,
1.88056380268556190092965814231, 4.08057500204924060580137561668, 5.38443518454005791519367248434, 8.228871898903973847729470185142, 9.689654350317624694751198454685, 12.03070897229269813180899973174, 13.04954048621681648300138715318, 14.49994084469257062017170752489, 15.79370636040615156474001498624, 16.75362060064551202608603531043