| L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s − 4·11-s + 3·13-s + 14-s + 16-s − 3·20-s − 4·22-s + 23-s + 4·25-s + 3·26-s + 28-s − 29-s − 2·31-s + 32-s − 3·35-s − 5·37-s − 3·40-s − 5·41-s − 7·43-s − 4·44-s + 46-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.670·20-s − 0.852·22-s + 0.208·23-s + 4/5·25-s + 0.588·26-s + 0.188·28-s − 0.185·29-s − 0.359·31-s + 0.176·32-s − 0.507·35-s − 0.821·37-s − 0.474·40-s − 0.780·41-s − 1.06·43-s − 0.603·44-s + 0.147·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−8.198536847044129409757551013399, −7.64063675716116306855148385477, −6.97470721465790476527130888674, −5.97542202954890271986707632396, −5.12244198907696155622145254954, −4.47010972768959845276328983230, −3.59280734014224177330155858430, −2.95501883207598714731421959025, −1.62271894495161954472460503089, 0,
1.62271894495161954472460503089, 2.95501883207598714731421959025, 3.59280734014224177330155858430, 4.47010972768959845276328983230, 5.12244198907696155622145254954, 5.97542202954890271986707632396, 6.97470721465790476527130888674, 7.64063675716116306855148385477, 8.198536847044129409757551013399
