| L(s) = 1 | + i·2-s + 0.470i·3-s − 4-s − 0.470·6-s + 3.30i·7-s − i·8-s + 2.77·9-s + 4.71·11-s − 0.470i·12-s + 2.24i·13-s − 3.30·14-s + 16-s − 0.719i·17-s + 2.77i·18-s + 0.470·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.271i·3-s − 0.5·4-s − 0.192·6-s + 1.25i·7-s − 0.353i·8-s + 0.926·9-s + 1.42·11-s − 0.135i·12-s + 0.623i·13-s − 0.884·14-s + 0.250·16-s − 0.174i·17-s + 0.654i·18-s + 0.107·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.857391063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.857391063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.470iT - 3T^{2} \) |
| 7 | \( 1 - 3.30iT - 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 - 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 0.719iT - 17T^{2} \) |
| 19 | \( 1 - 0.470T + 19T^{2} \) |
| 23 | \( 1 + 5.80iT - 23T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.02iT - 37T^{2} \) |
| 41 | \( 1 + 0.0275T + 41T^{2} \) |
| 43 | \( 1 - 6.61iT - 43T^{2} \) |
| 47 | \( 1 + 4.33iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 8.80iT - 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 - 3.64iT - 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 5.21iT - 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−9.525209690490941585628960012109, −8.969114488423919043545351100765, −8.334087351930139724945395491740, −7.18905186590075328513343598623, −6.49046431242130203962152469515, −5.86100379111253045066972607068, −4.64247614638952412323079602108, −4.18465263110509622299301245166, −2.79542467900995834716840911017, −1.39737108339364922699609699868,
0.899860982485450110058481782859, 1.66503505717602899396904290183, 3.27432813408693497642219305645, 4.02042723822584557214702077800, 4.71947617032199971105949779778, 6.06225247526865146226502419058, 6.95338592867462637030328623385, 7.59524346571613622746048371625, 8.506366788508655311491799616509, 9.593690556796491334870377839441
