L(s) = 1 | + (−1.28 − 0.591i)2-s + (2.70 + 0.538i)3-s + (1.30 + 1.51i)4-s + (−2.42 + 1.61i)5-s + (−3.16 − 2.29i)6-s + (0.747 − 1.11i)7-s + (−0.770 − 2.72i)8-s + (4.27 + 1.77i)9-s + (4.07 − 0.646i)10-s + (−0.611 − 3.07i)11-s + (2.70 + 4.81i)12-s + (−3.24 − 3.24i)13-s + (−1.62 + 0.994i)14-s + (−7.43 + 3.08i)15-s + (−0.619 + 3.95i)16-s + (−2.21 + 3.47i)17-s + ⋯ |
L(s) = 1 | + (−0.908 − 0.418i)2-s + (1.56 + 0.311i)3-s + (0.650 + 0.759i)4-s + (−1.08 + 0.724i)5-s + (−1.29 − 0.936i)6-s + (0.282 − 0.422i)7-s + (−0.272 − 0.962i)8-s + (1.42 + 0.590i)9-s + (1.28 − 0.204i)10-s + (−0.184 − 0.926i)11-s + (0.780 + 1.39i)12-s + (−0.899 − 0.899i)13-s + (−0.433 + 0.265i)14-s + (−1.92 + 0.795i)15-s + (−0.154 + 0.987i)16-s + (−0.538 + 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.837745 - 0.000104788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837745 - 0.000104788i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 + 0.591i)T \) |
| 17 | \( 1 + (2.21 - 3.47i)T \) |
good | 3 | \( 1 + (-2.70 - 0.538i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (2.42 - 1.61i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.747 + 1.11i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.611 + 3.07i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (3.24 + 3.24i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.100 - 0.242i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.31 + 0.459i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.242 - 0.362i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (1.10 - 5.55i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.474 - 2.38i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.44 - 0.965i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-3.12 + 7.55i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.35 + 3.35i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.62 + 1.50i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (2.31 + 0.959i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.61 - 6.90i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + (-14.4 - 2.88i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-7.57 + 5.06i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.02 - 5.14i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (3.34 - 1.38i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (12.4 - 12.4i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.63 + 5.43i)T + (-37.1 + 89.6i)T^{2} \) |
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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
−15.08684697042183981547970649798, −13.85811321664866410742759137790, −12.51051142859785127222336597177, −11.02803143760538359481091481985, −10.28105214862178449077801003434, −8.821330685246306799222068538958, −8.018042599328517050455131638381, −7.21089217710812819114632852509, −3.81889295629028106353643813941, −2.82310425180528839490970442001,
2.29554687694537868636751235645, 4.60779877438163359313006357805, 7.19048829630718043210383559358, 7.84422844189414552263270301106, 8.931266869090699035417234770427, 9.553028695512526292189985880222, 11.49738733007547493229035294285, 12.56301504957501336104519169879, 14.07117032209123437130472977660, 15.04519620701946872733931564168