| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 28·7-s − 8·8-s − 10·10-s + 36·11-s + 13·13-s + 56·14-s + 16·16-s − 42·17-s − 112·19-s + 20·20-s − 72·22-s + 168·23-s + 25·25-s − 26·26-s − 112·28-s + 210·29-s − 76·31-s − 32·32-s + 84·34-s − 140·35-s + 278·37-s + 224·38-s − 40·40-s − 150·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.986·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.599·17-s − 1.35·19-s + 0.223·20-s − 0.697·22-s + 1.52·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.34·29-s − 0.440·31-s − 0.176·32-s + 0.423·34-s − 0.676·35-s + 1.23·37-s + 0.956·38-s − 0.158·40-s − 0.571·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 112 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 76 T + p^{3} T^{2} \) |
| 37 | \( 1 - 278 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 - 264 T + p^{3} T^{2} \) |
| 53 | \( 1 + 582 T + p^{3} T^{2} \) |
| 59 | \( 1 - 204 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 304 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 + 934 T + p^{3} T^{2} \) |
| 79 | \( 1 - 128 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 834 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1582 T + p^{3} T^{2} \) |
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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
−8.967703707821825967339304024986, −8.569400458122270947236770338348, −7.11923241279948658693927532741, −6.52428533070693877914546250544, −6.07256406442528688368345244449, −4.60948519166404568924984688747, −3.44609919432754347062215261757, −2.54530681633179669426886075689, −1.24184857881976474913033084527, 0,
1.24184857881976474913033084527, 2.54530681633179669426886075689, 3.44609919432754347062215261757, 4.60948519166404568924984688747, 6.07256406442528688368345244449, 6.52428533070693877914546250544, 7.11923241279948658693927532741, 8.569400458122270947236770338348, 8.967703707821825967339304024986
