| L(s) = 1 | + 4.07·5-s + 1.30i·7-s − 4.01i·11-s − 4.82i·13-s − 2.18·17-s + (−3.91 − 1.91i)19-s − 3.65i·23-s + 11.5·25-s + 9.24i·29-s + 4.74·31-s + 5.29i·35-s − 0.423i·37-s − 3.88i·41-s − 4.97i·43-s − 4.57i·47-s + ⋯ |
| L(s) = 1 | + 1.82·5-s + 0.491i·7-s − 1.21i·11-s − 1.33i·13-s − 0.530·17-s + (−0.898 − 0.438i)19-s − 0.762i·23-s + 2.31·25-s + 1.71i·29-s + 0.852·31-s + 0.895i·35-s − 0.0696i·37-s − 0.606i·41-s − 0.758i·43-s − 0.667i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.569623344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569623344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.91 + 1.91i)T \) |
| good | 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 1.30iT - 7T^{2} \) |
| 11 | \( 1 + 4.01iT - 11T^{2} \) |
| 13 | \( 1 + 4.82iT - 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 23 | \( 1 + 3.65iT - 23T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 0.423iT - 37T^{2} \) |
| 41 | \( 1 + 3.88iT - 41T^{2} \) |
| 43 | \( 1 + 4.97iT - 43T^{2} \) |
| 47 | \( 1 + 4.57iT - 47T^{2} \) |
| 53 | \( 1 - 1.30iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 5.08T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.16iT - 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−8.453855498261995734129003506125, −7.07934455088151239434567939655, −6.45512496160418772034085557056, −5.71644404083770890694421785859, −5.46603650737761456322151993027, −4.53984681647997786764302810805, −3.17401484328505426172016463476, −2.65291299675178631083001673002, −1.78676740949893943358474121362, −0.62564746484865738881834535584,
1.35706991014818621386439571293, 2.02692869977795321635176448044, 2.64092306757434830597145532623, 4.16450806035201515726335475997, 4.53796501517412144958877516901, 5.49158162233676216764116771497, 6.39844169530357666509193196742, 6.56245312453650289155007120713, 7.47906440580817503884369960006, 8.362605746411002435111594372978
