L(s) = 1 | − 3-s + 1.68i·5-s + 9-s − 3.53i·11-s + 2.93i·13-s − 1.68i·15-s − 2.01i·17-s − 1.69·19-s + 1.59i·23-s + 2.16·25-s − 27-s + 7.94·29-s + 4.95·31-s + 3.53i·33-s − 10.4·37-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.753i·5-s + 0.333·9-s − 1.06i·11-s + 0.812i·13-s − 0.434i·15-s − 0.487i·17-s − 0.389·19-s + 0.333i·23-s + 0.432·25-s − 0.192·27-s + 1.47·29-s + 0.889·31-s + 0.614i·33-s − 1.72·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.382811506\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.382811506\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 1.68iT - 5T^{2} \) |
| 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 2.01iT - 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.59iT - 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 - 4.95T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.86iT - 41T^{2} \) |
| 43 | \( 1 + 11.7iT - 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 - 2.92T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 61 | \( 1 - 12.5iT - 61T^{2} \) |
| 67 | \( 1 + 1.70iT - 67T^{2} \) |
| 71 | \( 1 - 6.13iT - 71T^{2} \) |
| 73 | \( 1 - 2.43iT - 73T^{2} \) |
| 79 | \( 1 + 0.865iT - 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.82iT - 97T^{2} \) |
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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
−8.879426002075205963761922485936, −8.478631111139727145419330047844, −7.19898599252760299791556349851, −6.80129342788800461598252988662, −6.01122664011015647593980740755, −5.20570525634957808808059484641, −4.24046740596716383162275130331, −3.29351568949929367155183880791, −2.34443700951146668747852898021, −0.884724762832102362827502359295,
0.70692367281668501987435423426, 1.86178744440819269884510125320, 3.11355135082364531220275109574, 4.42749461323607686833829634183, 4.80743799192691930191397391408, 5.72272157290186797910993125831, 6.55939153286595255714504251194, 7.29656646847929009862318163489, 8.287834144342790724064578595786, 8.742282642936319298417180198382