| L(s) = 1 | + (1.68 − 1.08i)2-s + (0.426 − 2.96i)3-s + (1.66 − 3.63i)4-s + (16.0 − 4.71i)5-s + (−2.49 − 5.45i)6-s + (17.1 + 19.7i)7-s + (−1.13 − 7.91i)8-s + (−8.63 − 2.53i)9-s + (21.9 − 25.3i)10-s + (2.91 + 1.87i)11-s + (−10.0 − 6.48i)12-s + (−15.3 + 17.6i)13-s + (50.2 + 14.7i)14-s + (−7.15 − 49.7i)15-s + (−10.4 − 12.0i)16-s + (−29.3 − 64.2i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (1.43 − 0.422i)5-s + (−0.169 − 0.371i)6-s + (0.926 + 1.06i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.693 − 0.800i)10-s + (0.0798 + 0.0513i)11-s + (−0.242 − 0.156i)12-s + (−0.327 + 0.377i)13-s + (0.959 + 0.281i)14-s + (−0.123 − 0.856i)15-s + (−0.163 − 0.188i)16-s + (−0.418 − 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.56719 - 1.54757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56719 - 1.54757i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 + 1.08i)T \) |
| 3 | \( 1 + (-0.426 + 2.96i)T \) |
| 23 | \( 1 + (80.0 + 75.9i)T \) |
| good | 5 | \( 1 + (-16.0 + 4.71i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (-17.1 - 19.7i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (-2.91 - 1.87i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (15.3 - 17.6i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (29.3 + 64.2i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (2.36 - 5.18i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-48.6 - 106. i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-14.9 - 103. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (369. + 108. i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (324. - 95.2i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (31.1 - 216. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-389. - 449. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-81.6 + 94.1i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (83.4 + 580. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (587. - 377. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-118. + 75.8i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (265. - 580. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-572. + 660. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-1.02e3 - 301. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (225. - 1.57e3i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (558. - 164. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−12.43256302927186563178429533665, −11.92457629093581425336654845713, −10.61053198296985118556057788627, −9.361869101293874920176700606259, −8.547057765453851240357809979302, −6.81751986822928233986385880951, −5.65385023003951671819491596484, −4.87331714346208035350126560703, −2.50839917835684383741928393564, −1.63671750343127539229820962756,
2.01372285356549819354482400408, 3.80713812410867244497200360952, 5.08165791723220426800236563128, 6.12107386920489536086170632075, 7.36862143346094180490467127135, 8.643525939184809751343029632282, 10.12500266249817623013929155017, 10.57802627896553434625121344674, 11.87829654831272361750246973633, 13.49853618634949596604554937490
