L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (0.707 + 0.707i)27-s − i·29-s + (−1.22 − 1.22i)33-s − 1.73·39-s + (−0.707 − 0.707i)47-s + 1.00i·49-s + 1.73i·51-s + (−1.22 + 1.22i)77-s + 1.73·79-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (0.707 + 0.707i)27-s − i·29-s + (−1.22 − 1.22i)33-s − 1.73·39-s + (−0.707 − 0.707i)47-s + 1.00i·49-s + 1.73i·51-s + (−1.22 + 1.22i)77-s + 1.73·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.039733441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039733441\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 17 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
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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
−8.457714101745182311596134380096, −8.021977937738477020945823597870, −7.32005853374415551431877449578, −6.46702021834482134418619937992, −5.83531502530942062233693456176, −4.75046478529581237819290956807, −3.62034092322816225368270861473, −2.97692268308369254140041407019, −2.06274917084108562298222236802, −0.55571625777988804915144207697,
2.12147644115869156985132922621, 2.61312002864699451883070136959, 3.71894925508174478877660590025, 4.69406246344594031177084865426, 4.96762859446513177932784349973, 6.56096352245600570858893375333, 6.86572215074744672041760449997, 7.74076859082581687566816730865, 8.982922602447847920837393861042, 9.281691803123108543652593144632