L(s) = 1 | + 3.30i·3-s + (−1.88 + 1.20i)5-s − 7.93·9-s − 5.53·11-s + 1.73i·13-s + (−3.97 − 6.23i)15-s + 3.73i·17-s − 2.64·19-s + 4.05i·23-s + (2.10 − 4.53i)25-s − 16.3i·27-s + 3.81·29-s + 9.80·31-s − 18.2i·33-s − 3.34i·37-s + ⋯ |
L(s) = 1 | + 1.90i·3-s + (−0.843 + 0.537i)5-s − 2.64·9-s − 1.66·11-s + 0.482i·13-s + (−1.02 − 1.60i)15-s + 0.904i·17-s − 0.605·19-s + 0.845i·23-s + (0.421 − 0.906i)25-s − 3.14i·27-s + 0.709·29-s + 1.76·31-s − 3.18i·33-s − 0.549i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2578245797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2578245797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.88 - 1.20i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.30iT - 3T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 3.73iT - 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.05iT - 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 9.80T + 31T^{2} \) |
| 37 | \( 1 + 3.34iT - 37T^{2} \) |
| 41 | \( 1 - 3.39T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 - 1.31iT - 47T^{2} \) |
| 53 | \( 1 + 6.60iT - 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 4.94iT - 67T^{2} \) |
| 71 | \( 1 + 7.95T + 71T^{2} \) |
| 73 | \( 1 + 4.39iT - 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 - 5.87iT - 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 2.85iT - 97T^{2} \) |
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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
−10.03307834033925210461122478225, −9.182250168441662256831777037611, −8.184573540118124573327628898065, −7.947417550227816425655682199339, −6.51134971063185844026140239444, −5.66374196294710451862521311849, −4.68921566301065240953729792501, −4.25231996592618193121807429565, −3.25474054909477664065935928674, −2.61312480200152314740709994432,
0.11123784611142949943337800129, 0.973053661454637670953070096734, 2.45272860971536109905615845959, 2.96603649424496607303575857294, 4.61078804843287435613304908883, 5.40616076614249419686217499970, 6.29895156510106384247001237001, 7.17908472555974312045376273378, 7.71090154210237785808981994531, 8.337887055525502219905933594234