| L(s) = 1 | − 4.58i·5-s + (3.81 − 5.87i)7-s − 2.74i·11-s + 23.8·13-s + 3.77·17-s − 22.4i·19-s − 6.70·23-s + 3.99·25-s + 30.2·29-s + 11.6·31-s + (−26.9 − 17.4i)35-s − 1.51i·37-s − 3.87·41-s + 39.3·43-s + 86.1i·47-s + ⋯ |
| L(s) = 1 | − 0.916i·5-s + (0.544 − 0.838i)7-s − 0.249i·11-s + 1.83·13-s + 0.221·17-s − 1.18i·19-s − 0.291·23-s + 0.159·25-s + 1.04·29-s + 0.377·31-s + (−0.768 − 0.499i)35-s − 0.0408i·37-s − 0.0945·41-s + 0.915·43-s + 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0322 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0322 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.558322100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558322100\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-3.81 + 5.87i)T \) |
| good | 5 | \( 1 + 4.58iT - 25T^{2} \) |
| 11 | \( 1 + 2.74iT - 121T^{2} \) |
| 13 | \( 1 - 23.8T + 169T^{2} \) |
| 17 | \( 1 - 3.77T + 289T^{2} \) |
| 19 | \( 1 + 22.4iT - 361T^{2} \) |
| 23 | \( 1 + 6.70T + 529T^{2} \) |
| 29 | \( 1 - 30.2T + 841T^{2} \) |
| 31 | \( 1 - 11.6T + 961T^{2} \) |
| 37 | \( 1 + 1.51iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 3.87T + 1.68e3T^{2} \) |
| 43 | \( 1 - 39.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 56.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 37.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 66.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 79.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 98.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 114. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 42.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 62.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 164.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 97.0iT - 9.40e3T^{2} \) |
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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
−8.595945933815037086411920246524, −8.259442836025513484183914523801, −7.25538538040809051150938541383, −6.38604111935327169092632011046, −5.53953928987851797902233141541, −4.57730551674679590680229037720, −4.03674939878370674889421327986, −2.88351928240287725631777887620, −1.33262671151310262253083651054, −0.78420775037948161751255073889,
1.24227762155571503881458230662, 2.29957724001532362866687756781, 3.30558571431027778607190550816, 4.10738798805128356811079384874, 5.31662211415514762926100919358, 6.06763335603557889573829962965, 6.64495348767865444735206426338, 7.70777945601053330850518558685, 8.418002919758234880904171712890, 8.958473171115138390253807810943
