| L(s) = 1 | + (−1.23 + 0.551i)2-s + (1.19 − 1.25i)3-s + (−0.105 + 0.117i)4-s + (−2.19 + 0.406i)5-s + (−0.787 + 2.21i)6-s + (−1.81 + 3.13i)7-s + (0.905 − 2.78i)8-s + (−0.150 − 2.99i)9-s + (2.50 − 1.71i)10-s + (−5.29 + 2.35i)11-s + (0.0212 + 0.273i)12-s + (−0.742 − 0.330i)13-s + (0.513 − 4.89i)14-s + (−2.11 + 3.24i)15-s + (0.382 + 3.63i)16-s + (0.527 − 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.876 + 0.390i)2-s + (0.689 − 0.724i)3-s + (−0.0529 + 0.0588i)4-s + (−0.983 + 0.181i)5-s + (−0.321 + 0.904i)6-s + (−0.684 + 1.18i)7-s + (0.320 − 0.984i)8-s + (−0.0501 − 0.998i)9-s + (0.790 − 0.543i)10-s + (−1.59 + 0.710i)11-s + (0.00612 + 0.0789i)12-s + (−0.206 − 0.0917i)13-s + (0.137 − 1.30i)14-s + (−0.545 + 0.837i)15-s + (0.0956 + 0.909i)16-s + (0.128 − 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0338604 + 0.221889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0338604 + 0.221889i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.19 + 1.25i)T \) |
| 5 | \( 1 + (2.19 - 0.406i)T \) |
| good | 2 | \( 1 + (1.23 - 0.551i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (1.81 - 3.13i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.29 - 2.35i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (0.742 + 0.330i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.527 + 1.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.00 - 6.15i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.197 - 1.88i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (7.33 + 1.55i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (1.47 - 0.314i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-2.11 + 1.53i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.60 + 1.60i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-4.09 + 7.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.8 - 2.31i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.97 - 9.16i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.50 - 1.11i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-1.04 + 0.467i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (3.12 - 0.664i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-2.75 - 8.49i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.96 + 5.05i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.47 - 0.737i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (6.17 + 6.85i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (0.00564 + 0.00410i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0365 - 0.00775i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.53139255087337644735814314954, −12.13535002290395069788254889571, −10.44333445120474923498931163834, −9.421212087802915294008579131813, −8.593248784161647080455875776898, −7.69452609035352457435285688429, −7.27059433322692043825761048810, −5.76615099989080463578879754308, −3.81802967394204623611283369228, −2.51414218267815869915180272249,
0.21676118957438190878552218056, 2.79632283065757746027790293298, 4.08874806013374511172351587592, 5.19318761362126712753596219382, 7.30417092779208033537227239064, 8.092420251576417134114067328539, 8.910112671022490480918130299694, 9.926023603442850559209793204272, 10.75790402423926019730558422992, 11.12762970556840315999907585051
