| L(s) = 1 | + (27.7 − 16i)2-s + (511. − 886. i)4-s + (110. + 191. i)5-s + (−3.92e4 + 2.08e4i)7-s − 3.27e4i·8-s + (6.11e3 + 3.53e3i)10-s + (−8.40e5 − 4.85e5i)11-s + 6.28e5i·13-s + (−7.55e5 + 1.20e6i)14-s + (−5.24e5 − 9.08e5i)16-s + (−3.28e6 + 5.68e6i)17-s + (6.62e6 − 3.82e6i)19-s + 2.26e5·20-s − 3.10e7·22-s + (2.62e7 − 1.51e7i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0157 + 0.0273i)5-s + (−0.883 + 0.468i)7-s − 0.353i·8-s + (0.0193 + 0.0111i)10-s + (−1.57 − 0.909i)11-s + 0.469i·13-s + (−0.375 + 0.599i)14-s + (−0.125 − 0.216i)16-s + (−0.560 + 0.970i)17-s + (0.614 − 0.354i)19-s + 0.0157·20-s − 1.28·22-s + (0.851 − 0.491i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00161i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.133046225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133046225\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-27.7 + 16i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (3.92e4 - 2.08e4i)T \) |
| good | 5 | \( 1 + (-110. - 191. i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (8.40e5 + 4.85e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 6.28e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (3.28e6 - 5.68e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-6.62e6 + 3.82e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.62e7 + 1.51e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 2.00e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (1.09e7 + 6.30e6i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-3.07e8 - 5.33e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 9.78e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.90e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.29e9 - 2.24e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.56e9 + 9.00e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.98e9 - 5.17e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.31e9 + 7.59e8i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.96e9 + 8.59e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.09e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.51e10 - 1.45e10i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-5.52e9 - 9.56e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.44e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.61e10 - 4.52e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.63e11iT - 7.15e21T^{2} \) |
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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
−11.19407687985606491218791132804, −10.46586400058217197660331686016, −9.277629512359500314507815191118, −8.136130871387013626942548640380, −6.63752257353994785208041934759, −5.75580432809755672916238506988, −4.60760897000868392930348240286, −3.14674592994335060789174798860, −2.45883839327373449851456559339, −0.74020668824231654094823137730,
0.52601664117650857069965356836, 2.41286037673614478308675787291, 3.37671629845900843919763826843, 4.77266952663312683312901778117, 5.65168728025976288640448988732, 7.10507274724252867765924136277, 7.61730449603801002681434474429, 9.235118391059398346191601331545, 10.24148482234038755341606358899, 11.28726999797694214496984036358
