| L(s) = 1 | + (0.948 + 1.83i)3-s + (4.15 − 0.397i)5-s + (−2.51 + 0.821i)7-s + (−0.743 + 1.04i)9-s + (−1.71 + 0.594i)11-s + (1.51 + 5.15i)13-s + (4.67 + 7.27i)15-s + (−1.20 − 0.482i)17-s + (−1.20 + 0.484i)19-s + (−3.89 − 3.84i)21-s + (4.64 − 1.19i)23-s + (12.2 − 2.35i)25-s + (3.52 + 0.506i)27-s + (−0.925 − 6.43i)29-s + (−8.98 + 0.428i)31-s + ⋯ |
| L(s) = 1 | + (0.547 + 1.06i)3-s + (1.85 − 0.177i)5-s + (−0.950 + 0.310i)7-s + (−0.247 + 0.347i)9-s + (−0.518 + 0.179i)11-s + (0.419 + 1.43i)13-s + (1.20 + 1.87i)15-s + (−0.292 − 0.117i)17-s + (−0.277 + 0.111i)19-s + (−0.850 − 0.839i)21-s + (0.968 − 0.248i)23-s + (2.44 − 0.471i)25-s + (0.677 + 0.0974i)27-s + (−0.171 − 1.19i)29-s + (−1.61 + 0.0768i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76983 + 1.26634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76983 + 1.26634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.51 - 0.821i)T \) |
| 23 | \( 1 + (-4.64 + 1.19i)T \) |
| good | 3 | \( 1 + (-0.948 - 1.83i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (-4.15 + 0.397i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (1.71 - 0.594i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-1.51 - 5.15i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.20 + 0.482i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (1.20 - 0.484i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.925 + 6.43i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (8.98 - 0.428i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-8.64 - 6.15i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (7.98 + 3.64i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.43 + 3.79i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.169 + 0.0976i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.57 + 4.79i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-5.99 - 1.45i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-1.23 - 0.636i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (0.261 + 1.35i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (5.41 + 6.24i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (6.43 - 8.18i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 1.31i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (4.32 + 9.47i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.149 - 3.14i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-1.75 + 3.84i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.26791320730573525740239459123, −9.823902116207886597376138359429, −9.116575356641661922664223077955, −8.769412653312938711475955409376, −6.91255839039653845964422981653, −6.19084024295992075787343699990, −5.24464726327912043204731407148, −4.21515842229978098733746802780, −2.94824223391939287322688600380, −1.95567546824787729136329684590,
1.23383249198057249014439442468, 2.46830261969518656250892658777, 3.20402384900418743404587418470, 5.25582123703494125220571538881, 6.01113363807217747171673199631, 6.81025373531635378902611741061, 7.60813344008582561085126231756, 8.762654746942811196779115055587, 9.454917684047713838990658040562, 10.42038239828143019782872065351
