L(s) = 1 | + 5-s − 4·11-s + 13-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 6·29-s − 4·31-s + 6·37-s + 2·41-s + 12·43-s − 8·47-s − 7·49-s − 14·53-s − 4·55-s − 12·59-s − 2·61-s + 65-s − 8·71-s − 2·73-s + 8·79-s − 12·83-s + 2·85-s − 6·89-s − 4·95-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s + 0.312·41-s + 1.82·43-s − 1.16·47-s − 49-s − 1.92·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.124·65-s − 0.949·71-s − 0.234·73-s + 0.900·79-s − 1.31·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.984189477416458754672061917520, −7.34689845150244954575088779748, −6.23186765922244566574282930469, −5.94629015825883792891429139086, −4.94284031943159442030500635338, −4.32039212045326519058636437187, −3.17969977983531980383290806892, −2.46257396202073637007313610811, −1.45821482974105581536616908885, 0,
1.45821482974105581536616908885, 2.46257396202073637007313610811, 3.17969977983531980383290806892, 4.32039212045326519058636437187, 4.94284031943159442030500635338, 5.94629015825883792891429139086, 6.23186765922244566574282930469, 7.34689845150244954575088779748, 7.984189477416458754672061917520