L(s) = 1 | + 1.41i·3-s + 1.41i·5-s − 7-s − 1.00·9-s + 1.41i·13-s − 2.00·15-s − 1.41i·19-s − 1.41i·21-s − 1.00·25-s − 1.41i·35-s − 2.00·39-s − 1.41i·45-s + 49-s + 2.00·57-s + 1.41i·59-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 1.41i·5-s − 7-s − 1.00·9-s + 1.41i·13-s − 2.00·15-s − 1.41i·19-s − 1.41i·21-s − 1.00·25-s − 1.41i·35-s − 2.00·39-s − 1.41i·45-s + 49-s + 2.00·57-s + 1.41i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8651038771\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8651038771\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 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
−9.735451752035746112398946584145, −9.433913439247296679071091604554, −8.607026814142739218987755268580, −7.13174827753211292367713663677, −6.79743285310743289059958710336, −5.91293476356486502448103500445, −4.77179641024740516438809012266, −3.96565929565125957551253785738, −3.22204258923318931906077651284, −2.42026367985485045517483883468,
0.65117120693111178818101502296, 1.65987128712138572885536344919, 2.91916260714663413256731236995, 4.00475312761389532604162435734, 5.30861785228837432171374603994, 5.87634861256748669143952084556, 6.67975245207948186020533603510, 7.68073789616821305850212967561, 8.134716525764504755790280969845, 8.874239992228947871295029907154