L(s) = 1 | + (−1.39 − 1.39i)3-s + (0.642 − 0.642i)7-s + 2.90i·9-s − 1.79·21-s + (1.26 + 1.26i)23-s + (2.65 − 2.65i)27-s + 1.61i·29-s + 1.17·41-s + (0.221 + 0.221i)43-s + (−0.221 + 0.221i)47-s + 0.175i·49-s + 1.90·61-s + (1.86 + 1.86i)63-s + (−1 + i)67-s − 3.52i·69-s + ⋯ |
L(s) = 1 | + (−1.39 − 1.39i)3-s + (0.642 − 0.642i)7-s + 2.90i·9-s − 1.79·21-s + (1.26 + 1.26i)23-s + (2.65 − 2.65i)27-s + 1.61i·29-s + 1.17·41-s + (0.221 + 0.221i)43-s + (−0.221 + 0.221i)47-s + 0.175i·49-s + 1.90·61-s + (1.86 + 1.86i)63-s + (−1 + i)67-s − 3.52i·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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8494281454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8494281454\) |
\(L(1)\) |
|
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 | \( 1 + (1.39 + 1.39i)T + iT^{2} \) |
| 7 | \( 1 + (-0.642 + 0.642i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 29 | \( 1 - 1.61iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 1.17T + T^{2} \) |
| 43 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 47 | \( 1 + (0.221 - 0.221i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.90T + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.642 - 0.642i)T + iT^{2} \) |
| 89 | \( 1 - 0.618iT - T^{2} \) |
| 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
−8.207231468710432141380127563183, −7.51316407908959689841889739193, −7.11056725553947279407585714631, −6.46457511320722973930393665498, −5.48766263249432347665791659882, −5.15061634596944888820653100741, −4.20446449558217589288754884532, −2.81215699025586221299169346496, −1.60281600933719855619694639227, −1.01571872114289678813820390500,
0.76714202379998934383389429875, 2.44225631067716770839916983976, 3.58241606877787143565565804641, 4.45774221869678381914768852804, 4.90628253044025022011121201068, 5.66082092459869383260244262741, 6.22797751513219079517244984083, 7.00146652905385253355984182672, 8.168295621737607452760806909314, 8.961784238820055236530000097575