| L(s) = 1 | + (32 − 55.4i)2-s + (888. + 1.53e3i)3-s + (−2.04e3 − 3.54e3i)4-s + (−1.30e4 + 2.26e4i)5-s + 1.13e5·6-s + (−2.18e5 + 2.21e5i)7-s − 2.62e5·8-s + (−7.80e5 + 1.35e6i)9-s + (8.36e5 + 1.44e6i)10-s + (−5.60e6 − 9.70e6i)11-s + (3.63e6 − 6.30e6i)12-s − 1.74e7·13-s + (5.27e6 + 1.92e7i)14-s − 4.64e7·15-s + (−8.38e6 + 1.45e7i)16-s + (8.48e7 + 1.46e8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.703 + 1.21i)3-s + (−0.249 − 0.433i)4-s + (−0.374 + 0.647i)5-s + 0.994·6-s + (−0.702 + 0.711i)7-s − 0.353·8-s + (−0.489 + 0.848i)9-s + (0.264 + 0.458i)10-s + (−0.953 − 1.65i)11-s + (0.351 − 0.609i)12-s − 1.00·13-s + (0.187 + 0.681i)14-s − 1.05·15-s + (−0.125 + 0.216i)16-s + (0.852 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.432589 + 1.15667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.432589 + 1.15667i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-32 + 55.4i)T \) |
| 7 | \( 1 + (2.18e5 - 2.21e5i)T \) |
| good | 3 | \( 1 + (-888. - 1.53e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (1.30e4 - 2.26e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (5.60e6 + 9.70e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 + 1.74e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-8.48e7 - 1.46e8i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (8.89e7 - 1.54e8i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-2.23e7 + 3.87e7i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + 3.14e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-1.48e9 - 2.57e9i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (2.86e9 - 4.95e9i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 2.70e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.32e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.66e10 - 2.88e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (1.01e11 + 1.76e11i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-1.44e11 - 2.50e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (1.05e11 - 1.82e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-2.96e11 - 5.13e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 - 1.43e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (3.23e11 + 5.59e11i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (9.00e11 - 1.55e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 + 4.71e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (1.31e12 - 2.27e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + 7.42e12T + 6.73e25T^{2} \) |
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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
−16.43388622465713292343745766069, −15.24653076033295346436950440535, −14.41370217236016873536729118902, −12.69294424539282499271592454768, −10.90252821858500600284056905147, −9.901364205720389963589310742795, −8.405063338646606783958106529243, −5.66307475700102631451840109785, −3.65368761205294173122189503424, −2.82379028073472069892657516563,
0.38537465632742703949773636713, 2.54738380356064688896284778271, 4.73934187318295763589757543889, 7.13429038853520218260124944497, 7.67018595561296950508619933936, 9.573000996607635198857941836500, 12.40808970555403097915242298567, 12.98923172588638749530221930047, 14.27013332106684044438589436433, 15.73113523464709785295052170028
