L(s) = 1 | + (0.382 − 0.923i)2-s + (1.17 + 0.785i)3-s + (−0.707 − 0.707i)4-s + (1.38 − 0.275i)5-s + (1.17 − 0.785i)6-s + (−0.923 + 0.382i)8-s + (0.382 + 0.923i)9-s + (0.275 − 1.38i)10-s + (−0.785 − 1.17i)11-s + (−0.275 − 1.38i)12-s + (1.84 + 0.765i)15-s + i·16-s + 0.999·18-s + (−1.17 − 0.785i)20-s + (−1.38 + 0.275i)22-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (1.17 + 0.785i)3-s + (−0.707 − 0.707i)4-s + (1.38 − 0.275i)5-s + (1.17 − 0.785i)6-s + (−0.923 + 0.382i)8-s + (0.382 + 0.923i)9-s + (0.275 − 1.38i)10-s + (−0.785 − 1.17i)11-s + (−0.275 − 1.38i)12-s + (1.84 + 0.765i)15-s + i·16-s + 0.999·18-s + (−1.17 − 0.785i)20-s + (−1.38 + 0.275i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.306372143\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.306372143\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-1.17 - 0.785i)T + (0.382 + 0.923i)T^{2} \) |
| 5 | \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \) |
| 7 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (0.785 + 1.17i)T + (-0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 29 | \( 1 + (-0.275 - 1.38i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 37 | \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.275 - 1.38i)T + (-0.923 - 0.382i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 97 | \( 1 + (0.923 - 0.382i)T^{2} \) |
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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
−9.103723854096230461502628607769, −8.792009756373349964182727723365, −7.941973546327452496632753182615, −6.45126355980711887035498171645, −5.55359206591662604701485106512, −5.05319523378446415595122120127, −3.97400967059642687903358709294, −3.07736970617101001118486075063, −2.55521073486427893356738208684, −1.45916208903649324265388859610,
1.88093291829976715953917198729, 2.50710377872369727617862136820, 3.46743140483254534398452339284, 4.74511586954859605815273635967, 5.47160172116228783670485803694, 6.44857568105983151013027757626, 6.96098753728167604331742835022, 7.74142881731795475517861371260, 8.328321751155888364407285402682, 9.135826696886198646918628365804