L(s) = 1 | + 3-s + 3.22i·5-s − i·7-s + 9-s + 5.74i·11-s + (3.60 − 0.0996i)13-s + 3.22i·15-s + 4.91·17-s + 1.28i·19-s − i·21-s + 4.39·23-s − 5.39·25-s + 27-s + 4.19·29-s + 6.68i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.44i·5-s − 0.377i·7-s + 0.333·9-s + 1.73i·11-s + (0.999 − 0.0276i)13-s + 0.832i·15-s + 1.19·17-s + 0.293i·19-s − 0.218i·21-s + 0.916·23-s − 1.07·25-s + 0.192·27-s + 0.779·29-s + 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0276 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0276 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.723667876\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.723667876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3.60 + 0.0996i)T \) |
good | 5 | \( 1 - 3.22iT - 5T^{2} \) |
| 11 | \( 1 - 5.74iT - 11T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 19 | \( 1 - 1.28iT - 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 6.68iT - 31T^{2} \) |
| 37 | \( 1 + 5.95iT - 37T^{2} \) |
| 41 | \( 1 + 9.82iT - 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 + 0.506iT - 47T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 - 2.70iT - 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 + 5.31iT - 67T^{2} \) |
| 71 | \( 1 - 2.25iT - 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 + 9.42iT - 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 9.36iT - 97T^{2} \) |
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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
−8.483795617582584095624971820888, −7.63177824046120232254347596847, −7.08583209812641135869824386254, −6.71605582734359602834199206507, −5.68279266480115045762169898132, −4.70979013457645245846688387125, −3.74501313783881484082598587207, −3.22372062154012519686055538017, −2.30971378703081680564548977606, −1.34995075441075808394433891654,
0.804030790202150393425481040144, 1.39047483801555103182149490615, 2.88482031003029753735817449909, 3.45917382159314916272556606080, 4.44646443893296401876985423889, 5.23771542741522476808046853774, 5.88347762271369652899228522450, 6.59048333938249824836689491109, 7.935678682190949629920838734923, 8.261124237395135017932957809500