L(s) = 1 | + 1.56i·3-s + i·7-s + 0.561·9-s + 6.12·11-s + 2i·13-s − 1.56i·17-s + 3.56·19-s − 1.56·21-s − 1.43i·23-s + 5.56i·27-s − 3.43·29-s + 9.12·31-s + 9.56i·33-s − 8.80i·37-s − 3.12·39-s + ⋯ |
L(s) = 1 | + 0.901i·3-s + 0.377i·7-s + 0.187·9-s + 1.84·11-s + 0.554i·13-s − 0.378i·17-s + 0.817·19-s − 0.340·21-s − 0.299i·23-s + 1.07i·27-s − 0.638·29-s + 1.63·31-s + 1.66i·33-s − 1.44i·37-s − 0.500·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.307767200\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.307767200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.56iT - 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 1.43iT - 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 8.80iT - 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 + 6.56iT - 43T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 + 1.12iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.87iT - 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 6.43iT - 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 - 1.31iT - 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.971995925196835771291625163652, −8.541229205261290089063530543563, −7.18274115301482560539431984636, −6.78492745121218711230331013765, −5.75379142903300602547713940141, −4.96455660050966067516727673827, −4.02580068298830885842620489189, −3.66808647126897038840813217162, −2.31501067941884029106351264902, −1.11281497066095856328870657308,
1.00244876753282514339032836496, 1.54546765895884468657678162668, 2.92609117601338841310720152516, 3.88331654587880621317949068624, 4.65110496010331723723461068862, 5.84390331609604569277444602572, 6.57603444304151916800840967957, 7.01194564757859043766357645120, 7.910914727899275087312089858127, 8.442004490897878620438887965390