| 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
−10.87180086678363816173399677878, −9.563035638146124131932748313221, −9.410222476568338111689557678408, −8.344158235585086565289596157893, −7.33793668559091561026669853093, −5.80830857240061228730124684816, −4.82215325085576519121941953978, −3.77648053897508655789909313214, −2.81525855971105250692631026214, −1.43448763563063801430051703571,
1.16706516063890114208586551585, 3.27770600048873112644659243848, 4.18172595074228744398673983821, 5.73600247780716222741583556014, 6.14486677511507555709360285264, 7.43736195869986862319763093069, 8.097146718003495904001486230959, 8.746551789245079707099513530851, 9.820481040410483557613957533511, 10.60614867909580452479487561700
