L(s) = 1 | + 1.41·3-s − i·5-s + (0.707 − 0.707i)7-s + 1.00·9-s − 1.41i·15-s + (1.00 − 1.00i)21-s + 1.41i·23-s − 25-s − 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s − 1.00i·45-s + 1.41·47-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + 1.41·3-s − i·5-s + (0.707 − 0.707i)7-s + 1.00·9-s − 1.41i·15-s + (1.00 − 1.00i)21-s + 1.41i·23-s − 25-s − 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s − 1.00i·45-s + 1.41·47-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.975597385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.975597385\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.127467650661190345456698317869, −8.359376513667225449201013032334, −7.66679280756963260765118705011, −7.32444558415404015246482034962, −5.83920138823794587695111952648, −5.01204313542183171913037913994, −4.05178209043252604440803154498, −3.51556376065953967518053034108, −2.17493756776900941870296590072, −1.34666811078089520755033665187,
1.99226898323628789385560294789, 2.47430116350105595534480659430, 3.42050594435959414388165862574, 4.20407433506505993895151178118, 5.40423386802856022619193045079, 6.28011466778037113510506983890, 7.31424431941453007342579611298, 7.78384591556370904443434960001, 8.607460566491536003778470449083, 9.125909770087743401389697383895