L(s) = 1 | + (0.5 − 0.866i)2-s + (1.01 − 1.40i)3-s + (−0.499 − 0.866i)4-s + (−2.20 + 0.358i)5-s + (−0.707 − 1.58i)6-s + (−1.41 − 2.23i)7-s − 0.999·8-s + (−0.936 − 2.85i)9-s + (−0.792 + 2.09i)10-s + (−0.184 + 0.106i)11-s + (−1.72 − 0.178i)12-s + 6.70·13-s + (−2.64 + 0.106i)14-s + (−1.73 + 3.46i)15-s + (−0.5 + 0.866i)16-s + (2.73 − 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.586 − 0.809i)3-s + (−0.249 − 0.433i)4-s + (−0.987 + 0.160i)5-s + (−0.288 − 0.645i)6-s + (−0.534 − 0.845i)7-s − 0.353·8-s + (−0.312 − 0.950i)9-s + (−0.250 + 0.661i)10-s + (−0.0557 + 0.0321i)11-s + (−0.497 − 0.0514i)12-s + 1.85·13-s + (−0.706 + 0.0285i)14-s + (−0.448 + 0.893i)15-s + (−0.125 + 0.216i)16-s + (0.664 − 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.677047 - 1.21203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.677047 - 1.21203i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.01 + 1.40i)T \) |
| 5 | \( 1 + (2.20 - 0.358i)T \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
good | 11 | \( 1 + (0.184 - 0.106i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + (-2.73 + 1.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.23 - 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.23 - 5.60i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.02iT - 29T^{2} \) |
| 31 | \( 1 + (0.261 - 0.150i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.17 + 3.56i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 2.02iT - 43T^{2} \) |
| 47 | \( 1 + (6.71 + 3.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.45 - 4.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.61 - 2.66i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.02iT - 71T^{2} \) |
| 73 | \( 1 + (-5.99 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.261 - 0.452i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 + (-5.28 + 9.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−12.05515668562766106183378354567, −11.33512409042783808260954687055, −10.25022885049933900200909411970, −9.039983720036046196764603614708, −7.929594933994931799197753947238, −7.09609970211361310009885040475, −5.83641365758779433800041683510, −3.79794972152092256071524237913, −3.34415172810786937832327986379, −1.16244445341726846688226505292,
3.12331765039558873473381359124, 3.97727537982964476841197415187, 5.27067475168077068052606793568, 6.43864607302438388711236139355, 8.003625566549581412115945881534, 8.529426655806672517913008765650, 9.481375736976988384905983637630, 10.80896577770797141023521151821, 11.78986101842287916978993194882, 12.82759086550897068279016150878