L(s) = 1 | + (1.20 + 0.733i)2-s + 3-s + (0.924 + 1.77i)4-s + (−2.15 + 0.609i)5-s + (1.20 + 0.733i)6-s + (0.566 − 0.566i)7-s + (−0.181 + 2.82i)8-s + 9-s + (−3.04 − 0.840i)10-s + (3.64 + 3.64i)11-s + (0.924 + 1.77i)12-s − 2.74i·13-s + (1.10 − 0.269i)14-s + (−2.15 + 0.609i)15-s + (−2.28 + 3.28i)16-s + (2.08 − 2.08i)17-s + ⋯ |
L(s) = 1 | + (0.855 + 0.518i)2-s + 0.577·3-s + (0.462 + 0.886i)4-s + (−0.962 + 0.272i)5-s + (0.493 + 0.299i)6-s + (0.214 − 0.214i)7-s + (−0.0642 + 0.997i)8-s + 0.333·9-s + (−0.964 − 0.265i)10-s + (1.09 + 1.09i)11-s + (0.267 + 0.511i)12-s − 0.760i·13-s + (0.294 − 0.0721i)14-s + (−0.555 + 0.157i)15-s + (−0.572 + 0.820i)16-s + (0.505 − 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79665 + 1.07296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79665 + 1.07296i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.733i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2.15 - 0.609i)T \) |
good | 7 | \( 1 + (-0.566 + 0.566i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.64 - 3.64i)T + 11iT^{2} \) |
| 13 | \( 1 + 2.74iT - 13T^{2} \) |
| 17 | \( 1 + (-2.08 + 2.08i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.79 + 5.79i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.28 + 4.28i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.63 - 2.63i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.10iT - 31T^{2} \) |
| 37 | \( 1 - 2.28iT - 37T^{2} \) |
| 41 | \( 1 - 2.27iT - 41T^{2} \) |
| 43 | \( 1 - 3.06iT - 43T^{2} \) |
| 47 | \( 1 + (-1.80 - 1.80i)T + 47iT^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + (5.56 - 5.56i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.82 - 4.82i)T + 61iT^{2} \) |
| 67 | \( 1 + 3.34iT - 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + (1.02 - 1.02i)T - 97iT^{2} \) |
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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
−12.41353617093473533684857699175, −11.63283657733915805359325980220, −10.58681005151296999827977713672, −9.126121636647461984959116969995, −8.038941439345632543862791958844, −7.30549113584791895919042508402, −6.40199995912672957177954498942, −4.63460141381413886454986577104, −3.97796958605964130647705380411, −2.58285792517752135436710432922,
1.69500213047663931395373284405, 3.62434216172141681033870091683, 4.04121773829139574727379628272, 5.65269915901481498528479726604, 6.78821567145435689455222543818, 8.164287813352658027465965596063, 8.982275788574276984064636241722, 10.25229122634434576755368434862, 11.33891124898287851103162258908, 12.00319049270623813607908411811