| L(s) = 1 | + (0.0411 + 1.41i)2-s + (0.755 − 0.654i)3-s + (−1.99 + 0.116i)4-s + (0.243 − 0.0350i)5-s + (0.956 + 1.04i)6-s + (0.255 − 0.558i)7-s + (−0.246 − 2.81i)8-s + (0.142 − 0.989i)9-s + (0.0596 + 0.343i)10-s + (0.432 − 1.47i)11-s + (−1.43 + 1.39i)12-s + (4.14 − 1.89i)13-s + (0.800 + 0.337i)14-s + (0.161 − 0.186i)15-s + (3.97 − 0.464i)16-s + (3.53 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (0.0291 + 0.999i)2-s + (0.436 − 0.378i)3-s + (−0.998 + 0.0582i)4-s + (0.109 − 0.0156i)5-s + (0.390 + 0.425i)6-s + (0.0964 − 0.211i)7-s + (−0.0872 − 0.996i)8-s + (0.0474 − 0.329i)9-s + (0.0188 + 0.108i)10-s + (0.130 − 0.444i)11-s + (−0.413 + 0.402i)12-s + (1.14 − 0.525i)13-s + (0.213 + 0.0902i)14-s + (0.0416 − 0.0480i)15-s + (0.993 − 0.116i)16-s + (0.857 + 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60887 + 0.432860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60887 + 0.432860i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0411 - 1.41i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (-0.525 + 4.76i)T \) |
| good | 5 | \( 1 + (-0.243 + 0.0350i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.255 + 0.558i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.432 + 1.47i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.14 + 1.89i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.53 - 2.27i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.18 - 4.94i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.15 + 8.01i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.68 - 5.40i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.07 - 0.442i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.368 - 2.55i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-9.01 + 7.81i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 + (4.92 + 2.24i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (7.90 - 3.60i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.44 - 1.25i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.37 - 8.08i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (9.41 - 2.76i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.44 + 1.57i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (6.56 + 14.3i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (5.49 + 0.790i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (6.41 + 7.40i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.02 - 14.0i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.60614867909580452479487561700, −9.820481040410483557613957533511, −8.746551789245079707099513530851, −8.097146718003495904001486230959, −7.43736195869986862319763093069, −6.14486677511507555709360285264, −5.73600247780716222741583556014, −4.18172595074228744398673983821, −3.27770600048873112644659243848, −1.16706516063890114208586551585,
1.43448763563063801430051703571, 2.81525855971105250692631026214, 3.77648053897508655789909313214, 4.82215325085576519121941953978, 5.80830857240061228730124684816, 7.33793668559091561026669853093, 8.344158235585086565289596157893, 9.410222476568338111689557678408, 9.563035638146124131932748313221, 10.87180086678363816173399677878
