| L(s) = 1 | + (2.69 − 1.32i)3-s − 8.37i·5-s − 7.27·7-s + (5.49 − 7.12i)9-s + 17.4·13-s + (−11.0 − 22.5i)15-s − 19.9i·17-s + 20.5·19-s + (−19.6 + 9.63i)21-s − 31.1i·23-s − 45.2·25-s + (5.37 − 26.4i)27-s + 45.5i·29-s − 8.48·31-s + 60.9i·35-s + ⋯ |
| L(s) = 1 | + (0.897 − 0.441i)3-s − 1.67i·5-s − 1.03·7-s + (0.610 − 0.791i)9-s + 1.34·13-s + (−0.739 − 1.50i)15-s − 1.17i·17-s + 1.07·19-s + (−0.933 + 0.458i)21-s − 1.35i·23-s − 1.80·25-s + (0.199 − 0.979i)27-s + 1.57i·29-s − 0.273·31-s + 1.74i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.356112671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356112671\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.69 + 1.32i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 8.37iT - 25T^{2} \) |
| 7 | \( 1 + 7.27T + 49T^{2} \) |
| 13 | \( 1 - 17.4T + 169T^{2} \) |
| 17 | \( 1 + 19.9iT - 289T^{2} \) |
| 19 | \( 1 - 20.5T + 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 - 45.5iT - 841T^{2} \) |
| 31 | \( 1 + 8.48T + 961T^{2} \) |
| 37 | \( 1 - 26.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 26.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.57T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.84iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 80.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 81.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 86.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 105. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 56.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 45.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 5.58T + 9.40e3T^{2} \) |
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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
−8.937211251182375409113069062264, −8.464616669106263567640022643775, −7.47466682931994335232708894503, −6.63923164961598939733903959578, −5.67830742266898520260060332163, −4.69941363386687386177353121853, −3.71096837946244842271664235782, −2.86079228298298958748228944151, −1.41491940307560568883213914105, −0.60110571070774308170480483300,
1.73481085132354448555946540746, 3.08705685869769922806992203448, 3.31659379986469537579805299093, 4.20195799019062296322894431630, 5.94988395956141413155181457661, 6.29488822317785172480218493569, 7.45249058127982944901391937629, 7.87820813619217361579429912562, 9.070557498585086434848616639001, 9.715543315810857614290070690232
