| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 5·7-s + 8·8-s − 10·10-s + 35·11-s − 13·13-s + 10·14-s + 16·16-s − 23·17-s − 30·19-s − 20·20-s + 70·22-s − 63·23-s + 25·25-s − 26·26-s + 20·28-s + 190·29-s + 330·31-s + 32·32-s − 46·34-s − 25·35-s + 43·37-s − 60·38-s − 40·40-s + 473·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.269·7-s + 0.353·8-s − 0.316·10-s + 0.959·11-s − 0.277·13-s + 0.190·14-s + 1/4·16-s − 0.328·17-s − 0.362·19-s − 0.223·20-s + 0.678·22-s − 0.571·23-s + 1/5·25-s − 0.196·26-s + 0.134·28-s + 1.21·29-s + 1.91·31-s + 0.176·32-s − 0.232·34-s − 0.120·35-s + 0.191·37-s − 0.256·38-s − 0.158·40-s + 1.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.421569016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421569016\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 35 T + p^{3} T^{2} \) |
| 17 | \( 1 + 23 T + p^{3} T^{2} \) |
| 19 | \( 1 + 30 T + p^{3} T^{2} \) |
| 23 | \( 1 + 63 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 330 T + p^{3} T^{2} \) |
| 37 | \( 1 - 43 T + p^{3} T^{2} \) |
| 41 | \( 1 - 473 T + p^{3} T^{2} \) |
| 43 | \( 1 + 232 T + p^{3} T^{2} \) |
| 47 | \( 1 + 270 T + p^{3} T^{2} \) |
| 53 | \( 1 - 193 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 679 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 T + p^{3} T^{2} \) |
| 71 | \( 1 - 899 T + p^{3} T^{2} \) |
| 73 | \( 1 - 154 T + p^{3} T^{2} \) |
| 79 | \( 1 - 215 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1308 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1019 T + p^{3} T^{2} \) |
| 97 | \( 1 + 427 T + p^{3} T^{2} \) |
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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
−9.444927115985669725537379010374, −8.413446658832219220998681241500, −7.76528740604891627950221795823, −6.64024669740583859555633735918, −6.20256068736164001038857357783, −4.84297872378098766786735636734, −4.32593660688313220975477566166, −3.30240041026006542020435665480, −2.19538806140023195930077091805, −0.878109690123576024139466014385,
0.878109690123576024139466014385, 2.19538806140023195930077091805, 3.30240041026006542020435665480, 4.32593660688313220975477566166, 4.84297872378098766786735636734, 6.20256068736164001038857357783, 6.64024669740583859555633735918, 7.76528740604891627950221795823, 8.413446658832219220998681241500, 9.444927115985669725537379010374
