L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 3·13-s + 3·17-s + 19-s + 3·21-s + 9·23-s − 5·25-s + 5·27-s + 9·29-s + 6·31-s − 6·37-s + 3·39-s − 6·41-s + 8·43-s + 2·49-s − 3·51-s + 9·53-s − 57-s + 3·59-s − 6·61-s + 6·63-s − 5·67-s − 9·69-s − 11·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.832·13-s + 0.727·17-s + 0.229·19-s + 0.654·21-s + 1.87·23-s − 25-s + 0.962·27-s + 1.67·29-s + 1.07·31-s − 0.986·37-s + 0.480·39-s − 0.937·41-s + 1.21·43-s + 2/7·49-s − 0.420·51-s + 1.23·53-s − 0.132·57-s + 0.390·59-s − 0.768·61-s + 0.755·63-s − 0.610·67-s − 1.08·69-s − 1.28·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.85575465024369969680173807790, −6.96799153209878257205290502685, −6.53960768360003366744381815312, −5.66239072013242117757546798846, −5.15452372624600957749330244768, −4.23241203287815701217305650391, −3.02904154314331278527625648872, −2.77639508583334935586458739507, −1.10220313050988827907566783803, 0,
1.10220313050988827907566783803, 2.77639508583334935586458739507, 3.02904154314331278527625648872, 4.23241203287815701217305650391, 5.15452372624600957749330244768, 5.66239072013242117757546798846, 6.53960768360003366744381815312, 6.96799153209878257205290502685, 7.85575465024369969680173807790