L(s) = 1 | + (−1.61 + 0.618i)3-s + (1 − i)7-s + (2.23 − 2.00i)9-s + 4.47i·11-s + (3 + 3i)13-s + (2.23 + 2.23i)17-s + 2i·19-s + (−1 + 2.23i)21-s + (−2.23 + 2.23i)23-s + (−2.38 + 4.61i)27-s + 4.47·29-s + 4·31-s + (−2.76 − 7.23i)33-s + (3 − 3i)37-s + (−6.70 − 3i)39-s + ⋯ |
L(s) = 1 | + (−0.934 + 0.356i)3-s + (0.377 − 0.377i)7-s + (0.745 − 0.666i)9-s + 1.34i·11-s + (0.832 + 0.832i)13-s + (0.542 + 0.542i)17-s + 0.458i·19-s + (−0.218 + 0.487i)21-s + (−0.466 + 0.466i)23-s + (−0.458 + 0.888i)27-s + 0.830·29-s + 0.718·31-s + (−0.481 − 1.25i)33-s + (0.493 − 0.493i)37-s + (−1.07 − 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.910676 + 0.450308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910676 + 0.450308i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (2.23 - 2.23i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.70 + 6.70i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.23 + 2.23i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + (9 - 9i)T - 97iT^{2} \) |
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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
−11.91643183053885871324136451364, −10.89425681136525979435902672482, −10.15980494105412588721091290465, −9.304052915595038920909501277771, −7.928900008981773018156893847854, −6.88018142651264645346305697017, −5.93317061011583505420907459422, −4.69823729292359967523597413842, −3.88826997094273363745296248787, −1.58007197925693039768896021020,
0.975701264454769177292084140345, 2.99439973037375114555512246422, 4.68610918289573643177172169771, 5.76113911921327072589666841386, 6.41670149233872565732193823297, 7.84159630611829227813236471232, 8.536673609276060105715686709295, 9.954181573427288442211915892205, 10.92963868845701674432784492390, 11.50456948901217277781713717769