| 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
−15.04519620701946872733931564168, −14.07117032209123437130472977660, −12.56301504957501336104519169879, −11.49738733007547493229035294285, −9.553028695512526292189985880222, −8.931266869090699035417234770427, −7.84422844189414552263270301106, −7.19048829630718043210383559358, −4.60779877438163359313006357805, −2.29554687694537868636751235645,
2.82310425180528839490970442001, 3.81889295629028106353643813941, 7.21089217710812819114632852509, 8.018042599328517050455131638381, 8.821330685246306799222068538958, 10.28105214862178449077801003434, 11.02803143760538359481091481985, 12.51051142859785127222336597177, 13.85811321664866410742759137790, 15.08684697042183981547970649798
