| L(s) = 1 | − 0.246i·2-s + 0.801i·3-s + 1.93·4-s + 0.198·6-s + 1.69i·7-s − 0.972i·8-s + 2.35·9-s − 0.911·11-s + 1.55i·12-s − 1.55i·13-s + 0.417·14-s + 3.63·16-s + 5.29i·17-s − 0.582i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.174i·2-s + 0.462i·3-s + 0.969·4-s + 0.0808·6-s + 0.639i·7-s − 0.343i·8-s + 0.785·9-s − 0.274·11-s + 0.448i·12-s − 0.431i·13-s + 0.111·14-s + 0.909·16-s + 1.28i·17-s − 0.137i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78647 + 0.421730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78647 + 0.421730i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.246iT - 2T^{2} \) |
| 3 | \( 1 - 0.801iT - 3T^{2} \) |
| 7 | \( 1 - 1.69iT - 7T^{2} \) |
| 11 | \( 1 + 0.911T + 11T^{2} \) |
| 13 | \( 1 + 1.55iT - 13T^{2} \) |
| 17 | \( 1 - 5.29iT - 17T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.29iT - 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 + 7.31iT - 43T^{2} \) |
| 47 | \( 1 + 2.04iT - 47T^{2} \) |
| 53 | \( 1 - 2.70iT - 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 0.542T + 61T^{2} \) |
| 67 | \( 1 + 13.9iT - 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 2.80iT - 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 1.55iT - 97T^{2} \) |
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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
−10.79110011525208356287277658726, −10.45073969516149171339596275903, −9.466732944631054120326922515273, −8.345812505115379171283533712253, −7.42032486371290377713141188460, −6.38215956647713260363527253030, −5.49642235566107655898576830373, −4.17372664186543601072221257871, −2.99187210331832901766312486516, −1.72414094542852027542423773171,
1.34842145085615906267278507493, 2.67562431272716237997252064461, 4.08810349412195904599782072681, 5.41888710098161002255850956624, 6.54710866953308697317752253562, 7.39605794751050914753537769292, 7.68036344545625708313279109398, 9.255845549539205039919164160570, 10.11043113106370336394079481131, 11.07497015205686283123923829643
