L(s) = 1 | − 5·7-s − 7·13-s + 19-s − 5·25-s + 4·31-s − 37-s − 8·43-s + 18·49-s − 13·61-s − 11·67-s + 17·73-s + 13·79-s + 35·91-s + 5·97-s + 7·103-s + 2·109-s + ⋯ |
L(s) = 1 | − 1.88·7-s − 1.94·13-s + 0.229·19-s − 25-s + 0.718·31-s − 0.164·37-s − 1.21·43-s + 18/7·49-s − 1.66·61-s − 1.34·67-s + 1.98·73-s + 1.46·79-s + 3.66·91-s + 0.507·97-s + 0.689·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−10.40150154645553667557841727686, −9.772574318258794189724223366751, −9.220031324604069206185525205678, −7.78745335695197737171573573692, −6.91823062220437394787576967524, −6.08584189960381722201120484886, −4.87832652992641646209905365152, −3.51757494847066237517697434263, −2.49929826070171115553006367440, 0,
2.49929826070171115553006367440, 3.51757494847066237517697434263, 4.87832652992641646209905365152, 6.08584189960381722201120484886, 6.91823062220437394787576967524, 7.78745335695197737171573573692, 9.220031324604069206185525205678, 9.772574318258794189724223366751, 10.40150154645553667557841727686