L(s) = 1 | − 3-s + i·5-s − i·7-s + i·11-s − i·15-s + i·21-s − 25-s + 27-s + (−1 + i)31-s − i·33-s + 35-s + i·37-s − i·41-s + (−1 + i)43-s + i·47-s + ⋯ |
L(s) = 1 | − 3-s + i·5-s − i·7-s + i·11-s − i·15-s + i·21-s − 25-s + 27-s + (−1 + i)31-s − i·33-s + 35-s + i·37-s − i·41-s + (−1 + i)43-s + i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4659341180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4659341180\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 - iT \) |
good | 3 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−9.430521449211641066472587314416, −8.294977475313306523069875504684, −7.39054367692512863470855136988, −6.90815646013958289820178704103, −6.31884979598993498018886589014, −5.39139363290322717121176680772, −4.60889904539865313522161807224, −3.72426876535623464523948216503, −2.75346435835196292032892571803, −1.45177036608733505871314145623,
0.33469002145740885266890933020, 1.73761473197157885847033641072, 2.95433699839723432261760017877, 4.08938326542885783086393827808, 5.04313855943985713538836514320, 5.77658773910378469798921689824, 5.87397406091283897161096183959, 7.04092414068731602341612599756, 8.128680282000543430201327862612, 8.678138912802137436126231872316