L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 6·13-s + 16-s − 4·17-s − 8·19-s − 2·22-s + 6·26-s + 6·29-s + 8·31-s + 32-s − 4·34-s − 2·37-s − 8·38-s − 10·41-s + 4·43-s − 2·44-s + 8·47-s − 7·49-s + 6·52-s − 4·53-s + 6·58-s + 12·59-s − 10·61-s + 8·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 1.66·13-s + 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.426·22-s + 1.17·26-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.685·34-s − 0.328·37-s − 1.29·38-s − 1.56·41-s + 0.609·43-s − 0.301·44-s + 1.16·47-s − 49-s + 0.832·52-s − 0.549·53-s + 0.787·58-s + 1.56·59-s − 1.28·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.18444806783537, −13.69131524775269, −13.26533240403062, −13.08531173443438, −12.30382045436337, −11.96669876319353, −11.24546755739086, −10.87028119690757, −10.42565342164547, −10.07381093600222, −9.046942121611005, −8.646051300048414, −8.271505061715295, −7.724196554782251, −6.801902267039344, −6.430531214969858, −6.187452366653392, −5.416364646250109, −4.724409098308974, −4.326422315248516, −3.773191917785765, −3.055384001197538, −2.466162388029965, −1.822865211595160, −1.035370397215068, 0,
1.035370397215068, 1.822865211595160, 2.466162388029965, 3.055384001197538, 3.773191917785765, 4.326422315248516, 4.724409098308974, 5.416364646250109, 6.187452366653392, 6.430531214969858, 6.801902267039344, 7.724196554782251, 8.271505061715295, 8.646051300048414, 9.046942121611005, 10.07381093600222, 10.42565342164547, 10.87028119690757, 11.24546755739086, 11.96669876319353, 12.30382045436337, 13.08531173443438, 13.26533240403062, 13.69131524775269, 14.18444806783537