| L(s) = 1 | + (2.85 − 0.136i)2-s + (−2.90 − 4.30i)3-s + (0.196 − 0.0187i)4-s + (−0.0350 − 0.0334i)5-s + (−8.89 − 11.9i)6-s + (4.71 + 24.4i)7-s + (−22.1 + 3.17i)8-s + (−10.1 + 25.0i)9-s + (−0.104 − 0.0908i)10-s + (−14.4 + 7.45i)11-s + (−0.650 − 0.791i)12-s + (87.7 + 16.9i)13-s + (16.8 + 69.2i)14-s + (−0.0422 + 0.248i)15-s + (−64.3 + 12.4i)16-s + (24.8 + 54.4i)17-s + ⋯ |
| L(s) = 1 | + (1.01 − 0.0481i)2-s + (−0.558 − 0.829i)3-s + (0.0245 − 0.00234i)4-s + (−0.00313 − 0.00299i)5-s + (−0.605 − 0.811i)6-s + (0.254 + 1.32i)7-s + (−0.977 + 0.140i)8-s + (−0.375 + 0.926i)9-s + (−0.00331 − 0.00287i)10-s + (−0.396 + 0.204i)11-s + (−0.0156 − 0.0190i)12-s + (1.87 + 0.360i)13-s + (0.320 + 1.32i)14-s + (−0.000726 + 0.00427i)15-s + (−1.00 + 0.193i)16-s + (0.354 + 0.776i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42046 + 0.931276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42046 + 0.931276i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.90 + 4.30i)T \) |
| 23 | \( 1 + (97.4 + 51.7i)T \) |
| good | 2 | \( 1 + (-2.85 + 0.136i)T + (7.96 - 0.760i)T^{2} \) |
| 5 | \( 1 + (0.0350 + 0.0334i)T + (5.94 + 124. i)T^{2} \) |
| 7 | \( 1 + (-4.71 - 24.4i)T + (-318. + 127. i)T^{2} \) |
| 11 | \( 1 + (14.4 - 7.45i)T + (772. - 1.08e3i)T^{2} \) |
| 13 | \( 1 + (-87.7 - 16.9i)T + (2.03e3 + 816. i)T^{2} \) |
| 17 | \( 1 + (-24.8 - 54.4i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-34.6 - 15.8i)T + (4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (22.8 - 239. i)T + (-2.39e4 - 4.61e3i)T^{2} \) |
| 31 | \( 1 + (149. - 117. i)T + (7.02e3 - 2.89e4i)T^{2} \) |
| 37 | \( 1 + (105. - 360. i)T + (-4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (171. - 179. i)T + (-3.27e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-61.3 + 77.9i)T + (-1.87e4 - 7.72e4i)T^{2} \) |
| 47 | \( 1 + (-58.1 - 33.5i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (202. + 233. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (135. - 705. i)T + (-1.90e5 - 7.63e4i)T^{2} \) |
| 61 | \( 1 + (-300. + 751. i)T + (-1.64e5 - 1.56e5i)T^{2} \) |
| 67 | \( 1 + (105. - 205. i)T + (-1.74e5 - 2.44e5i)T^{2} \) |
| 71 | \( 1 + (262. + 408. i)T + (-1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-269. + 590. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (415. - 143. i)T + (3.87e5 - 3.04e5i)T^{2} \) |
| 83 | \( 1 + (-565. + 539. i)T + (2.72e4 - 5.71e5i)T^{2} \) |
| 89 | \( 1 + (53.3 - 370. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-1.10e3 - 267. i)T + (8.11e5 + 4.18e5i)T^{2} \) |
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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
−12.26189400131258917183264983239, −11.64521770444940908958124267689, −10.53992818753658092817212737861, −8.798662382155006902801971431743, −8.237355205046718932618791569371, −6.44968782294698457388414626355, −5.82950399925590510232173113499, −4.90158893328124658040655494165, −3.31493937209978667930449185731, −1.75336903819820560038121358943,
0.58062853071791659870882793646, 3.54555979331975968205212197448, 4.03493770028271542536058470593, 5.33956433193218500961633037036, 6.05468054475694390561461048076, 7.53208752451679307168516491688, 8.961200813214178529447532055722, 9.992126451329269629686018688599, 11.02410943971666467238280530518, 11.60700366629544827665955791807
