L(s) = 1 | + (−1.84 − 0.770i)2-s + (−2.78 + 1.12i)3-s + (2.81 + 2.84i)4-s + (3.86 − 3.17i)5-s + (5.99 + 0.0674i)6-s + (4.75 − 4.75i)7-s + (−2.99 − 7.41i)8-s + (6.46 − 6.25i)9-s + (−9.57 + 2.88i)10-s + 11.9·11-s + (−11.0 − 4.74i)12-s + (−4.22 + 4.22i)13-s + (−12.4 + 5.10i)14-s + (−7.16 + 13.1i)15-s + (−0.188 + 15.9i)16-s + (9.35 − 9.35i)17-s + ⋯ |
L(s) = 1 | + (−0.922 − 0.385i)2-s + (−0.927 + 0.375i)3-s + (0.702 + 0.711i)4-s + (0.772 − 0.635i)5-s + (0.999 + 0.0112i)6-s + (0.678 − 0.678i)7-s + (−0.374 − 0.927i)8-s + (0.718 − 0.695i)9-s + (−0.957 + 0.288i)10-s + 1.08·11-s + (−0.918 − 0.395i)12-s + (−0.324 + 0.324i)13-s + (−0.887 + 0.364i)14-s + (−0.477 + 0.878i)15-s + (−0.0117 + 0.999i)16-s + (0.550 − 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.694717 - 0.267871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.694717 - 0.267871i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.84 + 0.770i)T \) |
| 3 | \( 1 + (2.78 - 1.12i)T \) |
| 5 | \( 1 + (-3.86 + 3.17i)T \) |
good | 7 | \( 1 + (-4.75 + 4.75i)T - 49iT^{2} \) |
| 11 | \( 1 - 11.9T + 121T^{2} \) |
| 13 | \( 1 + (4.22 - 4.22i)T - 169iT^{2} \) |
| 17 | \( 1 + (-9.35 + 9.35i)T - 289iT^{2} \) |
| 19 | \( 1 - 1.48T + 361T^{2} \) |
| 23 | \( 1 + (11.6 - 11.6i)T - 529iT^{2} \) |
| 29 | \( 1 + 39.3T + 841T^{2} \) |
| 31 | \( 1 + 43.6iT - 961T^{2} \) |
| 37 | \( 1 + (-49.1 - 49.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 27.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-8.84 - 8.84i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (15.0 + 15.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 61.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 84.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (65.7 - 65.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 14.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-16.0 + 16.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 9.32T + 6.24e3T^{2} \) |
| 83 | \( 1 + (12.7 - 12.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-6.90 - 6.90i)T + 9.40e3iT^{2} \) |
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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
−14.89560130831574644853677724593, −13.36565748726733673155587336878, −11.99084305344021805001338293800, −11.29008096294566333213791522389, −9.948323585055286245297239179299, −9.284640083347564952955609913558, −7.55499385392636211904273534664, −6.08178404748233135287049599362, −4.31468332075551157875556238753, −1.28227796038914560325935160389,
1.80968155185674960950717824300, 5.46612867276007839140388153358, 6.38468017967639571489463213234, 7.62680158766522626404470769083, 9.188714460105212338432652189082, 10.41026492020859946672006613814, 11.32871115227362518385076357199, 12.41521075742851717161634919307, 14.22481896077334585843445012478, 15.00112980514837189452218952493