L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s + 0.999·10-s + (−2.5 − 4.33i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 2·17-s − 19-s + (−0.499 − 0.866i)20-s + (−2.5 + 4.33i)22-s + (0.5 − 0.866i)23-s + (2 + 3.46i)25-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s + 0.316·10-s + (−0.753 − 1.30i)11-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.485·17-s − 0.229·19-s + (−0.111 − 0.193i)20-s + (−0.533 + 0.923i)22-s + (0.104 − 0.180i)23-s + (0.400 + 0.692i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.105829224\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105829224\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (8 + 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.722551025457789726846914988637, −8.820431277888768831931558527740, −8.086678215711343290644210372615, −7.44772717243905492103760467807, −6.17424350685833532434394312595, −5.40594257258444688788536376082, −4.16375786701929451733323131838, −3.15815482586006477424790153876, −2.32704008276974662909946581123, −0.64691438952779580472167653723,
1.15265776637237791357158142894, 2.61764318624072097055462757229, 4.19364425682168426822832850646, 4.86724183069534869203422497547, 5.75921637312770979383939824971, 6.94638715705133832647102307968, 7.46983869278417207048881754803, 8.307770088887986882928349069310, 9.027126650352188801344842607865, 10.09249153676200250276119746691