L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 2·11-s + 13-s − 3·14-s + 16-s + 3·17-s − 19-s − 2·22-s − 23-s + 26-s − 3·28-s + 5·29-s − 8·31-s + 32-s + 3·34-s + 2·37-s − 38-s + 8·41-s − 4·43-s − 2·44-s − 46-s + 8·47-s + 2·49-s + 52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 0.603·11-s + 0.277·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.426·22-s − 0.208·23-s + 0.196·26-s − 0.566·28-s + 0.928·29-s − 1.43·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.162·38-s + 1.24·41-s − 0.609·43-s − 0.301·44-s − 0.147·46-s + 1.16·47-s + 2/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.38449138200409532872488206121, −6.59963058966096832452859213522, −5.97144427239650363907851612810, −5.49496912411842582687962149484, −4.56919662708677370242021422062, −3.83863559471832068300431952075, −3.10405081527717178572117889752, −2.53602562641872553806958066275, −1.34782691664996410999357640926, 0,
1.34782691664996410999357640926, 2.53602562641872553806958066275, 3.10405081527717178572117889752, 3.83863559471832068300431952075, 4.56919662708677370242021422062, 5.49496912411842582687962149484, 5.97144427239650363907851612810, 6.59963058966096832452859213522, 7.38449138200409532872488206121