L(s) = 1 | + (0.580 − 0.814i)3-s + (−0.142 − 0.989i)4-s + (0.327 − 0.945i)7-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)12-s + (−0.0748 + 0.0588i)13-s + (−0.959 + 0.281i)16-s + (0.462 + 1.90i)19-s + (−0.580 − 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.959 − 0.281i)27-s + (−0.981 − 0.189i)28-s + (0.0395 − 0.0865i)31-s + (−0.888 + 0.458i)36-s + (1.91 − 0.182i)37-s + ⋯ |
L(s) = 1 | + (0.580 − 0.814i)3-s + (−0.142 − 0.989i)4-s + (0.327 − 0.945i)7-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)12-s + (−0.0748 + 0.0588i)13-s + (−0.959 + 0.281i)16-s + (0.462 + 1.90i)19-s + (−0.580 − 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.959 − 0.281i)27-s + (−0.981 − 0.189i)28-s + (0.0395 − 0.0865i)31-s + (−0.888 + 0.458i)36-s + (1.91 − 0.182i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.220446068\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.220446068\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.580 + 0.814i)T \) |
| 397 | \( 1 - T \) |
good | 2 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 5 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 7 | \( 1 + (-0.327 + 0.945i)T + (-0.786 - 0.618i)T^{2} \) |
| 11 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 13 | \( 1 + (0.0748 - 0.0588i)T + (0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.462 - 1.90i)T + (-0.888 + 0.458i)T^{2} \) |
| 23 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 29 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 31 | \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 + (-1.91 + 0.182i)T + (0.981 - 0.189i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 61 | \( 1 + (0.911 + 0.717i)T + (0.235 + 0.971i)T^{2} \) |
| 67 | \( 1 + (-0.283 + 0.0270i)T + (0.981 - 0.189i)T^{2} \) |
| 71 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (1.13 + 1.08i)T + (0.0475 + 0.998i)T^{2} \) |
| 79 | \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 97 | \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \) |
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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
−9.722028658825292574160184295645, −8.943358905000993443916764762447, −7.80676045280159486188483311061, −7.47275688362598724981326175182, −6.28862529516681926569165039767, −5.73282031352582731348754988621, −4.43999893747186323487394692408, −3.51758672381126258711255093582, −2.02155377616556610873458066062, −1.10735571530196574360001190970,
2.43355284283967069753834984978, 2.94311811709556211047621386382, 4.19652317818262626403729510528, 4.81862480835454314647451386125, 5.86668328184424545076229908873, 7.17273517179699607554783596854, 7.964630146595182198287927128339, 8.676253111734098882844858580041, 9.222715222712257517327285225224, 9.945676331787800107038827420620