L(s) = 1 | + (1.67 − 0.437i)3-s + (−2.23 + 1.41i)7-s + (2.61 − 1.46i)9-s − 3.83i·11-s + 5.99i·13-s + 2.07·17-s + 5.11i·19-s + (−3.12 + 3.34i)21-s + 8.57i·23-s + (3.74 − 3.59i)27-s + 5.64i·29-s − 1.95i·31-s + (−1.67 − 6.42i)33-s − 2.23·37-s + (2.61 + 10.0i)39-s + ⋯ |
L(s) = 1 | + (0.967 − 0.252i)3-s + (−0.845 + 0.534i)7-s + (0.872 − 0.488i)9-s − 1.15i·11-s + 1.66i·13-s + 0.502·17-s + 1.17i·19-s + (−0.682 + 0.730i)21-s + 1.78i·23-s + (0.721 − 0.692i)27-s + 1.04i·29-s − 0.351i·31-s + (−0.291 − 1.11i)33-s − 0.367·37-s + (0.419 + 1.60i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.204067194\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.204067194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.67 + 0.437i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
good | 11 | \( 1 + 3.83iT - 11T^{2} \) |
| 13 | \( 1 - 5.99iT - 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 5.11iT - 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 - 5.64iT - 29T^{2} \) |
| 31 | \( 1 + 1.95iT - 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 4.37iT - 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 2.02iT - 71T^{2} \) |
| 73 | \( 1 + 1.41iT - 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 2.86T + 89T^{2} \) |
| 97 | \( 1 - 0.206iT - 97T^{2} \) |
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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
−9.054679392080837613269780579231, −8.660545447346088472744970894325, −7.66266902844422643210944199271, −6.95672066357786494304753879515, −6.13284426775624131058690034829, −5.37110440846447955256924578494, −3.81514376272512034760179855314, −3.54014889619920440192687208742, −2.39038045459763245623405339015, −1.35643516374835466018576620026,
0.72545926956986723202313995982, 2.42001891366634289242324129435, 2.99590506385661080316138447894, 4.05879440028223405415788498291, 4.73367879438020720210974668339, 5.84456061747493075064496102648, 6.97555615973833482084633859791, 7.40686809561392238425926617009, 8.266812050861767040155365783275, 9.019303216033122450186492067788