L(s) = 1 | + (−1.22 − 0.707i)5-s + (−1.22 + 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 1.41i·41-s − 43-s + (1.22 + 0.707i)47-s + 2·55-s + (−1.22 − 0.707i)65-s + (0.5 + 0.866i)67-s + 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)5-s + (−1.22 + 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 1.41i·41-s − 43-s + (1.22 + 0.707i)47-s + 2·55-s + (−1.22 − 0.707i)65-s + (0.5 + 0.866i)67-s + 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5762696665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5762696665\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.611570057777741821033017916687, −8.196933830183564652033934561007, −7.68297674420443987053660628460, −6.83045016045643918950848124808, −5.84169771051646778767065579917, −5.05481438898857624395573915695, −4.26081058188177777243893952061, −3.67350046840134224942105899317, −2.54349786270367631372483893525, −1.26395969672897782674459178465,
0.35933448142704463429978873713, 2.15193637122187343174648184778, 3.26495586219787160874483694853, 3.64116853052654115677327253668, 4.71808483062269148402882622766, 5.56070801828993697873976343942, 6.43320689727932498710165543849, 7.24549952664142577611807395226, 7.73431400371598179443485331954, 8.606882260665573496659626337379