L(s) = 1 | − 6·5-s − 7·7-s + 36·11-s + 62·13-s − 114·17-s + 76·19-s − 24·23-s − 89·25-s − 54·29-s + 112·31-s + 42·35-s − 178·37-s − 378·41-s + 172·43-s − 192·47-s + 49·49-s + 402·53-s − 216·55-s + 396·59-s + 254·61-s − 372·65-s + 1.01e3·67-s + 840·71-s + 890·73-s − 252·77-s − 80·79-s − 108·83-s + ⋯ |
L(s) = 1 | − 0.536·5-s − 0.377·7-s + 0.986·11-s + 1.32·13-s − 1.62·17-s + 0.917·19-s − 0.217·23-s − 0.711·25-s − 0.345·29-s + 0.648·31-s + 0.202·35-s − 0.790·37-s − 1.43·41-s + 0.609·43-s − 0.595·47-s + 1/7·49-s + 1.04·53-s − 0.529·55-s + 0.873·59-s + 0.533·61-s − 0.709·65-s + 1.84·67-s + 1.40·71-s + 1.42·73-s − 0.372·77-s − 0.113·79-s − 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.763769769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763769769\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 254 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 80 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1010 T + p^{3} T^{2} \) |
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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
−9.462721951179014409369429757499, −8.759532632335360220724735244522, −8.046420865894661170600076981122, −6.85035290503471266410248357350, −6.40799379737590035443009070372, −5.24858383371421549637235267853, −4.00937605471572853659508677316, −3.53188403676786521693653133688, −2.02226335537543742977171335832, −0.71218789596236189887666092513,
0.71218789596236189887666092513, 2.02226335537543742977171335832, 3.53188403676786521693653133688, 4.00937605471572853659508677316, 5.24858383371421549637235267853, 6.40799379737590035443009070372, 6.85035290503471266410248357350, 8.046420865894661170600076981122, 8.759532632335360220724735244522, 9.462721951179014409369429757499