L(s) = 1 | + i·2-s − 4-s + 2.44·5-s − i·8-s + 2.44i·10-s − 4.24i·11-s + 0.717i·13-s + 16-s + 2.44·17-s + 4.89i·19-s − 2.44·20-s + 4.24·22-s + 6i·23-s + 0.999·25-s − 0.717·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.09·5-s − 0.353i·8-s + 0.774i·10-s − 1.27i·11-s + 0.198i·13-s + 0.250·16-s + 0.594·17-s + 1.12i·19-s − 0.547·20-s + 0.904·22-s + 1.25i·23-s + 0.199·25-s − 0.140·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.181335062\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.181335062\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 0.717iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 1.75iT - 29T^{2} \) |
| 31 | \( 1 + 9.08iT - 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 14.4iT - 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 4.18iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + 5.49iT - 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 3.16iT - 97T^{2} \) |
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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
−9.129094707079767184236864325563, −7.919038069461034333452410662656, −7.66255617908431191204016080495, −6.35664963141460217912142757774, −5.81967225802563471476035218669, −5.53580546354533380989977712440, −4.22257349122139931613206628372, −3.38029203801264367902914301147, −2.17892325390047760207801638332, −0.945017880390302524310348376166,
0.983601803966360018328431051460, 2.14359464356999346487198401635, 2.69886050586456376622402975167, 3.95172207021582565679731195503, 4.89071140050748373985903366585, 5.43197579863815911961606636034, 6.51981470837180984452625404234, 7.14685463526987785443411570130, 8.195290755794203630344517877799, 9.060545224264838445110168651274