L(s) = 1 | + (−0.707 − 0.707i)5-s + 2.05i·7-s + (−2.89 − 2.89i)11-s + (−0.887 + 0.887i)13-s + 7.70·17-s + (−1.96 + 1.96i)19-s − 1.75i·23-s + 1.00i·25-s + (1.03 − 1.03i)29-s − 1.03·31-s + (1.45 − 1.45i)35-s + (−7.76 − 7.76i)37-s + 1.08i·41-s + (4.29 + 4.29i)43-s + 8.19·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s + 0.776i·7-s + (−0.873 − 0.873i)11-s + (−0.246 + 0.246i)13-s + 1.86·17-s + (−0.450 + 0.450i)19-s − 0.365i·23-s + 0.200i·25-s + (0.192 − 0.192i)29-s − 0.186·31-s + (0.245 − 0.245i)35-s + (−1.27 − 1.27i)37-s + 0.170i·41-s + (0.654 + 0.654i)43-s + 1.19·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.529093416\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529093416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - 2.05iT - 7T^{2} \) |
| 11 | \( 1 + (2.89 + 2.89i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.887 - 0.887i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 19 | \( 1 + (1.96 - 1.96i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (-1.03 + 1.03i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 + (7.76 + 7.76i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.08iT - 41T^{2} \) |
| 43 | \( 1 + (-4.29 - 4.29i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + (-3.76 - 3.76i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.92 - 3.92i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.18 - 6.18i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.26 + 8.26i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.34iT - 71T^{2} \) |
| 73 | \( 1 + 14.3iT - 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + (-2.72 + 2.72i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.96iT - 89T^{2} \) |
| 97 | \( 1 - 7.11T + 97T^{2} \) |
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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
−8.806973478176059165930108995819, −7.85944223430460178038879663837, −7.56642438260420171599451135529, −6.24438613192719733478549774529, −5.62073243489541710910855259551, −5.03929126893200243042911239741, −3.88468851514034888302247181257, −3.07522229283411289717843918898, −2.10597637105804058172511617992, −0.68632375520969044746550166299,
0.834958301549164591381508047464, 2.20373687647700818345961039645, 3.23160888946845774675394445879, 3.98683469800385627346638997885, 5.00002838482431200266657500793, 5.57928856329531114527759883613, 6.80896777414346911319401815649, 7.33519604594065351626986922134, 7.88256608187153924309786790918, 8.684067553383742108010102450342