L(s) = 1 | + (0.437 − 0.437i)2-s + (−0.156 − 0.987i)3-s + 0.618i·4-s + (−0.5 − 0.363i)6-s + (1.34 + 1.34i)7-s + (0.707 + 0.707i)8-s + (−0.951 + 0.309i)9-s + (0.610 − 0.0966i)12-s + 1.17·14-s + (−1.14 + 1.14i)17-s + (−0.280 + 0.550i)18-s + (1.11 − 1.53i)21-s + (0.587 − 0.809i)24-s + (0.453 + 0.891i)27-s + (−0.831 + 0.831i)28-s + ⋯ |
L(s) = 1 | + (0.437 − 0.437i)2-s + (−0.156 − 0.987i)3-s + 0.618i·4-s + (−0.5 − 0.363i)6-s + (1.34 + 1.34i)7-s + (0.707 + 0.707i)8-s + (−0.951 + 0.309i)9-s + (0.610 − 0.0966i)12-s + 1.17·14-s + (−1.14 + 1.14i)17-s + (−0.280 + 0.550i)18-s + (1.11 − 1.53i)21-s + (0.587 − 0.809i)24-s + (0.453 + 0.891i)27-s + (−0.831 + 0.831i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.628600308\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.628600308\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.437 + 0.437i)T - iT^{2} \) |
| 7 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.831 + 0.831i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.437 - 0.437i)T + iT^{2} \) |
| 59 | \( 1 - 1.90T + T^{2} \) |
| 61 | \( 1 + 0.618T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.17iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.61iT - T^{2} \) |
| 83 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 89 | \( 1 - 1.90T + T^{2} \) |
| 97 | \( 1 + (1.34 + 1.34i)T + iT^{2} \) |
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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
−8.595382019210247336689902244595, −8.119556697054733332049403907025, −7.47370294908503586289997736354, −6.52807469001902937513128031578, −5.70395858326445545434851456118, −5.02080830628307596754767032367, −4.22360934256000167968923296062, −3.04956610437760816675615802331, −2.10168817394705730961119653307, −1.78079392583461774626607168626,
0.849682442915890559267681909081, 2.16112232666304044760086889760, 3.65620238927538959336813042364, 4.30899210923829249236715384919, 4.99702037935592366430710959523, 5.26210139776893847388740152377, 6.53586135381433801840079338471, 7.01757110894645813145241761738, 7.915693729251065993446480145118, 8.719392627471101124702051358258