L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 6·11-s − 2·13-s − 2·14-s + 16-s + 2·20-s + 6·22-s + 23-s − 25-s − 2·26-s − 2·28-s − 6·29-s + 8·31-s + 32-s − 4·35-s + 2·40-s − 10·41-s − 12·43-s + 6·44-s + 46-s + 8·47-s − 3·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 1.80·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.447·20-s + 1.27·22-s + 0.208·23-s − 1/5·25-s − 0.392·26-s − 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.676·35-s + 0.316·40-s − 1.56·41-s − 1.82·43-s + 0.904·44-s + 0.147·46-s + 1.16·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.326716093\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.326716093\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.54257875774255250348856283685, −10.12888984830209574104614830759, −9.630000082494789242014648422170, −8.623377758533207484865735274631, −7.04614505207963284764458800400, −6.43907907295841169043019574490, −5.55267882834237934594942614166, −4.27895780918103049245443906167, −3.16921099094622245450045571331, −1.72081981979020341015639926740,
1.72081981979020341015639926740, 3.16921099094622245450045571331, 4.27895780918103049245443906167, 5.55267882834237934594942614166, 6.43907907295841169043019574490, 7.04614505207963284764458800400, 8.623377758533207484865735274631, 9.630000082494789242014648422170, 10.12888984830209574104614830759, 11.54257875774255250348856283685