| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 7·7-s + 8·8-s + 10·10-s − 46·11-s − 8·13-s + 14·14-s + 16·16-s + 79·17-s − 121·19-s + 20·20-s − 92·22-s − 49·23-s + 25·25-s − 16·26-s + 28·28-s − 132·29-s − 5·31-s + 32·32-s + 158·34-s + 35·35-s − 38·37-s − 242·38-s + 40·40-s − 26·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.26·11-s − 0.170·13-s + 0.267·14-s + 1/4·16-s + 1.12·17-s − 1.46·19-s + 0.223·20-s − 0.891·22-s − 0.444·23-s + 1/5·25-s − 0.120·26-s + 0.188·28-s − 0.845·29-s − 0.0289·31-s + 0.176·32-s + 0.796·34-s + 0.169·35-s − 0.168·37-s − 1.03·38-s + 0.158·40-s − 0.0990·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 46 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 79 T + p^{3} T^{2} \) |
| 19 | \( 1 + 121 T + p^{3} T^{2} \) |
| 23 | \( 1 + 49 T + p^{3} T^{2} \) |
| 29 | \( 1 + 132 T + p^{3} T^{2} \) |
| 31 | \( 1 + 5 T + p^{3} T^{2} \) |
| 37 | \( 1 + 38 T + p^{3} T^{2} \) |
| 41 | \( 1 + 26 T + p^{3} T^{2} \) |
| 43 | \( 1 - 114 T + p^{3} T^{2} \) |
| 47 | \( 1 + 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 609 T + p^{3} T^{2} \) |
| 59 | \( 1 - 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 635 T + p^{3} T^{2} \) |
| 67 | \( 1 - 30 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1090 T + p^{3} T^{2} \) |
| 73 | \( 1 + 338 T + p^{3} T^{2} \) |
| 79 | \( 1 - 477 T + p^{3} T^{2} \) |
| 83 | \( 1 + 579 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 + 920 T + p^{3} T^{2} \) |
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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
−8.173578167083621799567108152475, −7.78782128570704223389377207524, −6.73537216276838020380814712559, −5.88928414911087639111842295144, −5.23865389787050646198470824937, −4.48599341157825893326577463392, −3.40483778790959105059860365554, −2.45293053101606564165170099365, −1.60133604642208272494507909756, 0,
1.60133604642208272494507909756, 2.45293053101606564165170099365, 3.40483778790959105059860365554, 4.48599341157825893326577463392, 5.23865389787050646198470824937, 5.88928414911087639111842295144, 6.73537216276838020380814712559, 7.78782128570704223389377207524, 8.173578167083621799567108152475
