L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s + 3·11-s + 14-s + 16-s − 2·19-s − 3·20-s + 3·22-s − 6·23-s + 4·25-s + 28-s + 6·29-s − 5·31-s + 32-s − 3·35-s − 2·37-s − 2·38-s − 3·40-s + 6·41-s − 10·43-s + 3·44-s − 6·46-s − 6·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.904·11-s + 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.670·20-s + 0.639·22-s − 1.25·23-s + 4/5·25-s + 0.188·28-s + 1.11·29-s − 0.898·31-s + 0.176·32-s − 0.507·35-s − 0.328·37-s − 0.324·38-s − 0.474·40-s + 0.937·41-s − 1.52·43-s + 0.452·44-s − 0.884·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.41464292852056870921396123834, −6.59276603410744826558826615915, −6.14489361856934397455540154451, −5.07417183227043597911266205476, −4.51812797571223389780552821284, −3.82252581662889396750879050561, −3.42104357350238937241802492879, −2.28651529879700464449438957349, −1.33173223924086766131485229394, 0,
1.33173223924086766131485229394, 2.28651529879700464449438957349, 3.42104357350238937241802492879, 3.82252581662889396750879050561, 4.51812797571223389780552821284, 5.07417183227043597911266205476, 6.14489361856934397455540154451, 6.59276603410744826558826615915, 7.41464292852056870921396123834