L(s) = 1 | − 2·3-s + 9-s + 11-s − 2·13-s + 2·17-s + 4·19-s − 5·25-s + 4·27-s − 6·29-s + 2·31-s − 2·33-s − 2·37-s + 4·39-s − 6·41-s + 6·47-s − 4·51-s + 2·53-s − 8·57-s − 2·59-s + 2·61-s + 4·67-s + 6·73-s + 10·75-s − 12·79-s − 11·81-s + 4·83-s + 12·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 25-s + 0.769·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.875·47-s − 0.560·51-s + 0.274·53-s − 1.05·57-s − 0.260·59-s + 0.256·61-s + 0.488·67-s + 0.702·73-s + 1.15·75-s − 1.35·79-s − 1.22·81-s + 0.439·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−7.31791855213182380983081450907, −6.70447827827555092922258035081, −5.88775375513861432281170052200, −5.46271539420894129881926188297, −4.84097237982656894471238952292, −3.94671991002227840524447836866, −3.16004817019678492833397907382, −2.08161523286470591521829887950, −1.03961887724386371443510119306, 0,
1.03961887724386371443510119306, 2.08161523286470591521829887950, 3.16004817019678492833397907382, 3.94671991002227840524447836866, 4.84097237982656894471238952292, 5.46271539420894129881926188297, 5.88775375513861432281170052200, 6.70447827827555092922258035081, 7.31791855213182380983081450907