L(s) = 1 | + (−0.642 + 0.642i)3-s + (0.221 + 0.221i)7-s + 0.175i·9-s − 0.284·21-s + (−1.39 + 1.39i)23-s + (−0.754 − 0.754i)27-s + 0.618i·29-s + 1.90·41-s + (−1.26 + 1.26i)43-s + (1.26 + 1.26i)47-s − 0.902i·49-s − 1.17·61-s + (−0.0388 + 0.0388i)63-s + (−1 − i)67-s − 1.79i·69-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.642i)3-s + (0.221 + 0.221i)7-s + 0.175i·9-s − 0.284·21-s + (−1.39 + 1.39i)23-s + (−0.754 − 0.754i)27-s + 0.618i·29-s + 1.90·41-s + (−1.26 + 1.26i)43-s + (1.26 + 1.26i)47-s − 0.902i·49-s − 1.17·61-s + (−0.0388 + 0.0388i)63-s + (−1 − i)67-s − 1.79i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7562467498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7562467498\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.642 - 0.642i)T - iT^{2} \) |
| 7 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.39 - 1.39i)T - iT^{2} \) |
| 29 | \( 1 - 0.618iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.90T + T^{2} \) |
| 43 | \( 1 + (1.26 - 1.26i)T - iT^{2} \) |
| 47 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.17T + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 89 | \( 1 - 1.61iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.142443549184207396802143394328, −7.88747275125320847945565997013, −7.74597899820184922506942652841, −6.49734260490206989090093218222, −5.81237408284658537514129972172, −5.21428702350929557558206117393, −4.43673131063969231206036375133, −3.71386322009086328585138449273, −2.58471226218742250919164149738, −1.50060423787936564122638613176,
0.46283721215525029424093956826, 1.68229271237527879165761840344, 2.66940267718612434434992335927, 3.90247837241598044999420731036, 4.50918643023291850792708552031, 5.67871601829204475250348288373, 6.06464901266121416766938407234, 6.89878047868684388915888753419, 7.48098688825115870700739003556, 8.284931981271262806103262190644