| L(s) = 1 | + (1.68 − 1.08i)2-s + (−1.06 + 7.38i)3-s + (1.66 − 3.63i)4-s + (4.79 − 1.40i)5-s + (6.20 + 13.5i)6-s + (2.08 + 2.40i)7-s + (−1.13 − 7.91i)8-s + (−27.5 − 8.08i)9-s + (6.54 − 7.55i)10-s + (49.6 + 31.8i)11-s + (25.1 + 16.1i)12-s + (16.5 − 19.0i)13-s + (6.11 + 1.79i)14-s + (5.31 + 36.9i)15-s + (−10.4 − 12.0i)16-s + (21.3 + 46.6i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.204 + 1.42i)3-s + (0.207 − 0.454i)4-s + (0.429 − 0.125i)5-s + (0.421 + 0.923i)6-s + (0.112 + 0.130i)7-s + (−0.0503 − 0.349i)8-s + (−1.02 − 0.299i)9-s + (0.207 − 0.238i)10-s + (1.36 + 0.874i)11-s + (0.604 + 0.388i)12-s + (0.352 − 0.406i)13-s + (0.116 + 0.0342i)14-s + (0.0914 + 0.635i)15-s + (−0.163 − 0.188i)16-s + (0.303 + 0.665i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.11485 + 1.48071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11485 + 1.48071i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 + 1.08i)T \) |
| 5 | \( 1 + (-4.79 + 1.40i)T \) |
| 23 | \( 1 + (101. - 43.4i)T \) |
| good | 3 | \( 1 + (1.06 - 7.38i)T + (-25.9 - 7.60i)T^{2} \) |
| 7 | \( 1 + (-2.08 - 2.40i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (-49.6 - 31.8i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-16.5 + 19.0i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-21.3 - 46.6i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (49.6 - 108. i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (106. + 233. i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-34.8 - 242. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-309. - 90.9i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-121. + 35.6i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-1.31 + 9.16i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-325. - 375. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-421. + 486. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (72.9 + 507. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-389. + 250. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-643. + 413. i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (148. - 324. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-530. + 611. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (582. + 171. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (60.8 - 423. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (1.42e3 - 418. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−11.89844491103367066982268381097, −10.92131508162017495352282275145, −9.940937201488373922612891518758, −9.587112010926383564070311137667, −8.221655300558489789497248697345, −6.38961098686831484932749049148, −5.52386845432744432190883786360, −4.29787946710294378011482342433, −3.68453259963211904166732915791, −1.75702896234060440721526808327,
0.979412081921902176552066487037, 2.43815534262220175679989693040, 4.09120495175819944796905215315, 5.73277294490887072774585597523, 6.50926384886638690968678985876, 7.17010249156536066790117383844, 8.358924036922477082134647716468, 9.377754835794283563376731486294, 11.18681180717058822839728701095, 11.65350512996050756981766178749
