L(s) = 1 | − 1.73i·2-s + 2·3-s + 5·4-s − 1.73i·5-s − 3.46i·6-s − 13.8i·7-s − 22.5i·8-s − 23·9-s − 2.99·10-s − 13.8i·11-s + 10·12-s − 23.9·14-s − 3.46i·15-s + 1.00·16-s + 117·17-s + 39.8i·18-s + ⋯ |
L(s) = 1 | − 0.612i·2-s + 0.384·3-s + 0.625·4-s − 0.154i·5-s − 0.235i·6-s − 0.748i·7-s − 0.995i·8-s − 0.851·9-s − 0.0948·10-s − 0.379i·11-s + 0.240·12-s − 0.458·14-s − 0.0596i·15-s + 0.0156·16-s + 1.66·17-s + 0.521i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.26484 - 1.68162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26484 - 1.68162i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + 1.73iT - 8T^{2} \) |
| 3 | \( 1 - 2T + 27T^{2} \) |
| 5 | \( 1 + 1.73iT - 125T^{2} \) |
| 7 | \( 1 + 13.8iT - 343T^{2} \) |
| 11 | \( 1 + 13.8iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 117T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 78T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 143. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 271. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 104T + 7.95e4T^{2} \) |
| 47 | \( 1 - 301. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 93T + 1.48e5T^{2} \) |
| 59 | \( 1 + 284. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 145T + 2.26e5T^{2} \) |
| 67 | \( 1 - 786. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 458. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 789. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 976. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 200. iT - 9.12e5T^{2} \) |
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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
−11.90152420499075338523475973876, −11.05134804718197398552149884746, −10.22824045953998964852109891144, −9.089852905947676091212248003528, −7.87215039040839901917117402162, −6.85642439129705637202471357487, −5.48501557872537541854338693268, −3.70528307012963988231683941259, −2.69794993348018223144316428941, −0.945791144304933191258361455779,
2.07957064753648795519160772782, 3.35432841855299622498109349817, 5.46230330957167140991024366127, 6.10536104673807200527334911870, 7.58827406840953507209647140960, 8.208822071315627447775540158688, 9.439198943712432812725712764857, 10.62642806607817428357026077402, 11.79935189690261994538018618336, 12.37886404018165738373650293554