L(s) = 1 | + 1.73i·3-s − 3.46·5-s + 2i·7-s − 2.99·9-s + 3.46i·11-s − 5.99i·15-s − 3.46·21-s + 6.99·25-s − 5.19i·27-s + 10.3·29-s + 10i·31-s − 5.99·33-s − 6.92i·35-s + 10.3·45-s + 3·49-s + ⋯ |
L(s) = 1 | + 0.999i·3-s − 1.54·5-s + 0.755i·7-s − 0.999·9-s + 1.04i·11-s − 1.54i·15-s − 0.755·21-s + 1.39·25-s − 0.999i·27-s + 1.92·29-s + 1.79i·31-s − 1.04·33-s − 1.17i·35-s + 1.54·45-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.277839 + 0.670764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.277839 + 0.670764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 17.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−12.41369092813890677831441508406, −11.98363740071009398644598262747, −10.98133586445151051878938205998, −10.00823606997005476219059125758, −8.833813212233360580687419028768, −8.086044169057755940303815821901, −6.77976237236339813057152656198, −5.10839154095295853648707963912, −4.24584208548445000431373439403, −2.98832760465601501203027787079,
0.65769462898042245966475753634, 3.09169602061645601325033086587, 4.33600377406399389572265462396, 6.07907118619607722103108258081, 7.23662014052043436285253384095, 7.947438183672344558450776274376, 8.738128664291525707390370251644, 10.52509838180128325938893143389, 11.47764814622900285917972765520, 12.00100887687889700048252778414