L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 3·11-s − 13-s + 16-s − 3·17-s − 7·19-s − 3·20-s + 3·22-s + 9·23-s + 4·25-s − 26-s − 3·29-s + 8·31-s + 32-s − 3·34-s − 37-s − 7·38-s − 3·40-s − 3·41-s − 43-s + 3·44-s + 9·46-s + 4·50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.904·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 1.60·19-s − 0.670·20-s + 0.639·22-s + 1.87·23-s + 4/5·25-s − 0.196·26-s − 0.557·29-s + 1.43·31-s + 0.176·32-s − 0.514·34-s − 0.164·37-s − 1.13·38-s − 0.474·40-s − 0.468·41-s − 0.152·43-s + 0.452·44-s + 1.32·46-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.31158885394624733981936629724, −6.76082700816054387377402550505, −6.29645493082283472827089407499, −5.17635760841560625235601199895, −4.45293258684100283248097536087, −4.09227354691083565933218441269, −3.27048100202727774302178485328, −2.47731534992902817333063967817, −1.29823669942233208644881078784, 0,
1.29823669942233208644881078784, 2.47731534992902817333063967817, 3.27048100202727774302178485328, 4.09227354691083565933218441269, 4.45293258684100283248097536087, 5.17635760841560625235601199895, 6.29645493082283472827089407499, 6.76082700816054387377402550505, 7.31158885394624733981936629724