| L(s) = 1 | + (−2.74 − 10.2i)2-s + (−22.3 + 38.7i)3-s + (−41.8 + 24.1i)4-s + (−35.0 + 35.0i)5-s + (458. + 122. i)6-s + (−112. + 420. i)7-s + (−117. − 117. i)8-s + (−638. − 1.10e3i)9-s + (454. + 262. i)10-s + (−1.83e3 + 492. i)11-s − 2.16e3i·12-s + (2.15e3 − 410. i)13-s + 4.61e3·14-s + (−574. − 2.14e3i)15-s + (−2.42e3 + 4.20e3i)16-s + (2.97e3 − 1.71e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.342 − 1.27i)2-s + (−0.829 + 1.43i)3-s + (−0.653 + 0.377i)4-s + (−0.280 + 0.280i)5-s + (2.12 + 0.568i)6-s + (−0.328 + 1.22i)7-s + (−0.229 − 0.229i)8-s + (−0.876 − 1.51i)9-s + (0.454 + 0.262i)10-s + (−1.38 + 0.369i)11-s − 1.25i·12-s + (0.982 − 0.186i)13-s + 1.68·14-s + (−0.170 − 0.634i)15-s + (−0.592 + 1.02i)16-s + (0.605 − 0.349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.304056 + 0.353166i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304056 + 0.353166i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.15e3 + 410. i)T \) |
| good | 2 | \( 1 + (2.74 + 10.2i)T + (-55.4 + 32i)T^{2} \) |
| 3 | \( 1 + (22.3 - 38.7i)T + (-364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (35.0 - 35.0i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + (112. - 420. i)T + (-1.01e5 - 5.88e4i)T^{2} \) |
| 11 | \( 1 + (1.83e3 - 492. i)T + (1.53e6 - 8.85e5i)T^{2} \) |
| 17 | \( 1 + (-2.97e3 + 1.71e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.31e3 - 353. i)T + (4.07e7 + 2.35e7i)T^{2} \) |
| 23 | \( 1 + (-4.73e3 - 2.73e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-6.24e3 + 1.08e4i)T + (-2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (3.93e4 - 3.93e4i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (4.16e4 - 1.11e4i)T + (2.22e9 - 1.28e9i)T^{2} \) |
| 41 | \( 1 + (-9.28e3 - 3.46e4i)T + (-4.11e9 + 2.37e9i)T^{2} \) |
| 43 | \( 1 + (-8.07e4 + 4.66e4i)T + (3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-3.16e4 - 3.16e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + 1.62e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.51e4 + 5.64e4i)T + (-3.65e10 - 2.10e10i)T^{2} \) |
| 61 | \( 1 + (-9.81e4 - 1.70e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.11e4 + 1.16e5i)T + (-7.83e10 + 4.52e10i)T^{2} \) |
| 71 | \( 1 + (-4.72e4 - 1.26e4i)T + (1.10e11 + 6.40e10i)T^{2} \) |
| 73 | \( 1 + (-1.89e4 - 1.89e4i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + 9.10e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-2.40e5 + 2.40e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + (1.15e6 - 3.10e5i)T + (4.30e11 - 2.48e11i)T^{2} \) |
| 97 | \( 1 + (-4.69e5 - 1.25e5i)T + (7.21e11 + 4.16e11i)T^{2} \) |
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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
−18.91273740598438607334965106424, −17.95866244083797625393955362505, −15.94387847811095746641095494988, −15.37522226127315051786568723449, −12.54910472683845283680144181444, −11.29246482914916893539678452389, −10.38767000764850213085276152642, −9.157717977757484835211837689319, −5.48795353621795341263773439617, −3.22527062811828659237485880773,
0.43183964991025141328242473483, 5.78132330457806841212759422896, 7.13588796173315896874779311629, 8.093241306291882742182780888206, 10.96705166813043647160099514372, 12.74699061973102648556330287426, 13.87932205015604590121520768425, 16.02344422416308575376306776280, 16.82030646816184086152179064344, 17.95334278669009648480816954325
