L(s) = 1 | + (34.6 − 34.6i)2-s + (539. + 539. i)3-s + 1.68e3i·4-s + (1.29e4 − 8.80e3i)5-s + 3.74e4·6-s + (2.59e4 − 2.59e4i)7-s + (2.00e5 + 2.00e5i)8-s + 4.96e4i·9-s + (1.42e5 − 7.53e5i)10-s − 3.09e6·11-s + (−9.10e5 + 9.10e5i)12-s + (2.18e5 + 2.18e5i)13-s − 1.79e6i·14-s + (1.17e7 + 2.21e6i)15-s + 7.01e6·16-s + (−5.19e6 + 5.19e6i)17-s + ⋯ |
L(s) = 1 | + (0.542 − 0.542i)2-s + (0.739 + 0.739i)3-s + 0.412i·4-s + (0.826 − 0.563i)5-s + 0.801·6-s + (0.220 − 0.220i)7-s + (0.765 + 0.765i)8-s + 0.0934i·9-s + (0.142 − 0.753i)10-s − 1.74·11-s + (−0.304 + 0.304i)12-s + (0.0452 + 0.0452i)13-s − 0.238i·14-s + (1.02 + 0.194i)15-s + 0.417·16-s + (−0.215 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0446i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.999 - 0.0446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.54668 + 0.0569457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54668 + 0.0569457i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.29e4 + 8.80e3i)T \) |
good | 2 | \( 1 + (-34.6 + 34.6i)T - 4.09e3iT^{2} \) |
| 3 | \( 1 + (-539. - 539. i)T + 5.31e5iT^{2} \) |
| 7 | \( 1 + (-2.59e4 + 2.59e4i)T - 1.38e10iT^{2} \) |
| 11 | \( 1 + 3.09e6T + 3.13e12T^{2} \) |
| 13 | \( 1 + (-2.18e5 - 2.18e5i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 + (5.19e6 - 5.19e6i)T - 5.82e14iT^{2} \) |
| 19 | \( 1 + 4.05e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + (1.59e8 + 1.59e8i)T + 2.19e16iT^{2} \) |
| 29 | \( 1 - 8.83e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 8.39e8T + 7.87e17T^{2} \) |
| 37 | \( 1 + (2.38e9 - 2.38e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 - 3.94e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + (-2.54e9 - 2.54e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 + (-2.54e9 + 2.54e9i)T - 1.16e20iT^{2} \) |
| 53 | \( 1 + (8.59e8 + 8.59e8i)T + 4.91e20iT^{2} \) |
| 59 | \( 1 + 3.93e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 4.84e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + (4.84e10 - 4.84e10i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 + 7.15e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-3.85e10 - 3.85e10i)T + 2.29e22iT^{2} \) |
| 79 | \( 1 - 5.14e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (3.03e11 + 3.03e11i)T + 1.06e23iT^{2} \) |
| 89 | \( 1 - 5.82e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (-3.48e11 + 3.48e11i)T - 6.93e23iT^{2} \) |
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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
−20.92729481797513986628410183470, −20.33126884187733928892754928086, −17.75672369998011034146672458084, −15.98998276798011088847621612310, −14.02598161754478999503278145304, −12.75803637698671461821729296690, −10.40180966744711828180407884376, −8.457248597144657193926439775223, −4.70816182474164037004906422772, −2.64614412983406766514947750726,
2.16476082680819636060835882374, 5.67242890665179942033278325665, 7.63571855024723545752002536904, 10.24813640370536796506365414151, 13.27384290315460026992425211311, 14.10458887185281925422439377061, 15.62679497692445092579045736916, 18.11540059880151452199493998315, 19.19523277225996387298291287526, 21.09381555638808190060630407422