L(s) = 1 | + (−0.427 + 0.246i)2-s + (−0.243 + 0.908i)3-s + (−0.878 + 1.52i)4-s + (−0.284 + 2.21i)5-s + (−0.120 − 0.448i)6-s + (1.83 − 3.18i)7-s − 1.85i·8-s + (1.83 + 1.05i)9-s + (−0.426 − 1.01i)10-s + (−0.177 + 0.664i)11-s + (−1.16 − 1.16i)12-s + (−2.92 − 2.11i)13-s + 1.81i·14-s + (−1.94 − 0.798i)15-s + (−1.29 − 2.24i)16-s + (2.29 − 0.614i)17-s + ⋯ |
L(s) = 1 | + (−0.302 + 0.174i)2-s + (−0.140 + 0.524i)3-s + (−0.439 + 0.760i)4-s + (−0.127 + 0.991i)5-s + (−0.0490 − 0.183i)6-s + (0.694 − 1.20i)7-s − 0.655i·8-s + (0.610 + 0.352i)9-s + (−0.134 − 0.322i)10-s + (−0.0536 + 0.200i)11-s + (−0.337 − 0.337i)12-s + (−0.810 − 0.585i)13-s + 0.485i·14-s + (−0.502 − 0.206i)15-s + (−0.324 − 0.562i)16-s + (0.556 − 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.582322 + 0.436185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.582322 + 0.436185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.284 - 2.21i)T \) |
| 13 | \( 1 + (2.92 + 2.11i)T \) |
good | 2 | \( 1 + (0.427 - 0.246i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.243 - 0.908i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-1.83 + 3.18i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.177 - 0.664i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.29 + 0.614i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.29 + 1.41i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 0.350i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (8.24 - 4.75i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.81 + 4.81i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.917 + 1.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.534 + 0.143i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.560 + 2.09i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 3.80T + 47T^{2} \) |
| 53 | \( 1 + (2.47 + 2.47i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.69 + 10.0i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.09 - 5.36i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.6 - 6.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.73 + 6.47i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 3.37iT - 73T^{2} \) |
| 79 | \( 1 + 3.12iT - 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 + (-3.26 - 0.874i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-6.12 - 3.53i)T + (48.5 + 84.0i)T^{2} \) |
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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
−15.17111955510005159917186281021, −14.08758352591801375893395154295, −13.07731063234616716268420672656, −11.57666662090569415875560760990, −10.44599434715628388856834479832, −9.628496737169697969486661669173, −7.59292602498354729656286397160, −7.37566146090300836140953140618, −4.84683710638821427977951804903, −3.51417157787690166020860247077,
1.58077064638148416848972069651, 4.78967095557496256949660213614, 5.82239224600822766260727798930, 7.78749380709282734997390476197, 9.009175614638137520895738359459, 9.807832172286613526455916638719, 11.62453130695327603604065035541, 12.25906308352914227387036727239, 13.51587895856440888201125608648, 14.70923772711220539872615408647