| L(s) = 1 | + (0.987 + 0.156i)2-s + (−0.757 + 1.48i)3-s + (0.951 + 0.309i)4-s + (−1.65 + 1.50i)5-s + (−0.980 + 1.35i)6-s + (2.92 − 1.49i)7-s + (0.891 + 0.453i)8-s + (0.126 + 0.174i)9-s + (−1.86 + 1.23i)10-s + (−3.16 + 0.977i)11-s + (−1.18 + 1.18i)12-s + (0.966 − 6.10i)13-s + (3.12 − 1.01i)14-s + (−0.991 − 3.59i)15-s + (0.809 + 0.587i)16-s + (−0.839 − 5.30i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (−0.437 + 0.858i)3-s + (0.475 + 0.154i)4-s + (−0.738 + 0.674i)5-s + (−0.400 + 0.551i)6-s + (1.10 − 0.564i)7-s + (0.315 + 0.160i)8-s + (0.0421 + 0.0580i)9-s + (−0.590 + 0.389i)10-s + (−0.955 + 0.294i)11-s + (−0.340 + 0.340i)12-s + (0.268 − 1.69i)13-s + (0.835 − 0.271i)14-s + (−0.255 − 0.928i)15-s + (0.202 + 0.146i)16-s + (−0.203 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11972 + 0.631203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11972 + 0.631203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (1.65 - 1.50i)T \) |
| 11 | \( 1 + (3.16 - 0.977i)T \) |
| good | 3 | \( 1 + (0.757 - 1.48i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (-2.92 + 1.49i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.966 + 6.10i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.839 + 5.30i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.213 - 0.657i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.92 - 3.92i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.689 + 2.12i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.63 - 3.37i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 2.91i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.24 + 0.404i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.38 - 4.38i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.57 + 0.800i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (3.09 + 0.489i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-0.820 - 0.266i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (5.96 - 8.20i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-9.05 + 9.05i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.58 - 1.15i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.52 + 2.99i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (3.40 - 2.47i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.96 - 0.944i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 4.90iT - 89T^{2} \) |
| 97 | \( 1 + (-2.36 + 14.9i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.93650341111701684268838685005, −12.88603974758365372932935608781, −11.42697580480838151534914519699, −10.91704037888156678360950509475, −10.10227650633221455645632981559, −7.970899523165499033814635191443, −7.32678734916237936168582792045, −5.37181362382979242679307157454, −4.62574468669472318501453789795, −3.15837138396957380219109508102,
1.77489475736435608116868452676, 4.16881578371085812859953852198, 5.33632404386077576757167114756, 6.66683970917155865765946560685, 7.88844987544492690433321206907, 8.906795129136801808292207505157, 10.99459357759588402347774293689, 11.60366025389194577963238893358, 12.52243032640704333579310179125, 13.16346869972467068439053192340
