L(s) = 1 | + (1.93 + 1.93i)5-s − 1.41·7-s + (−0.732 + 0.732i)11-s + (1.73 + 1.73i)13-s + 5.27i·17-s + (5.27 − 5.27i)19-s − 3.46i·23-s + 2.46i·25-s + (−2.31 + 2.31i)29-s + 9.14i·31-s + (−2.73 − 2.73i)35-s + (−2.46 + 2.46i)37-s + 4.52·41-s + (−3.48 − 3.48i)43-s + 10.3·47-s + ⋯ |
L(s) = 1 | + (0.863 + 0.863i)5-s − 0.534·7-s + (−0.220 + 0.220i)11-s + (0.480 + 0.480i)13-s + 1.28i·17-s + (1.21 − 1.21i)19-s − 0.722i·23-s + 0.492i·25-s + (−0.429 + 0.429i)29-s + 1.64i·31-s + (−0.461 − 0.461i)35-s + (−0.405 + 0.405i)37-s + 0.705·41-s + (−0.531 − 0.531i)43-s + 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831929034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831929034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.93 - 1.93i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + (0.732 - 0.732i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.27iT - 17T^{2} \) |
| 19 | \( 1 + (-5.27 + 5.27i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + (2.31 - 2.31i)T - 29iT^{2} \) |
| 31 | \( 1 - 9.14iT - 31T^{2} \) |
| 37 | \( 1 + (2.46 - 2.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + (3.48 + 3.48i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + (-8.24 - 8.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (8.92 - 8.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (3 + 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.86 - 3.86i)T - 67iT^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 - 4.92iT - 73T^{2} \) |
| 79 | \( 1 + 2.17iT - 79T^{2} \) |
| 83 | \( 1 + (10.7 + 10.7i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.107081872132576809593596489526, −8.687251853684049132749733761744, −7.39087559931190881437880326323, −6.84380092748590356895160736918, −6.14942804001696691326178931183, −5.43253642572615104101886869065, −4.33569714929385400390549918940, −3.25289202111295630172314314666, −2.54730588489390712946377078666, −1.38206872757384166715666108205,
0.64846542355813443282390295689, 1.79385424372066582129563774812, 2.98361620450611989669930216169, 3.86920332335744862767420047085, 5.09449898201422336716745592930, 5.63455498177519021914977657318, 6.22780166571128846838904152407, 7.46662635502968484860783033346, 7.957758793224008447215579644392, 9.090723958866061429825992232795