| L(s) = 1 | − 2.58·3-s + 3.70·9-s + 4.70·11-s + 2.58·13-s + 1.81·17-s + 5.95·19-s − 7.40·23-s − 1.81·27-s − 6.70·29-s − 1.54·31-s − 12.1·33-s − 7.40·37-s − 6.70·39-s + 1.54·41-s − 4·43-s + 10.6·47-s − 4.70·51-s − 11.4·53-s − 15.4·57-s − 13.2·59-s + 2.85·61-s − 8·67-s + 19.1·69-s + 7.40·71-s − 12.4·73-s − 8.10·79-s − 6.40·81-s + ⋯ |
| L(s) = 1 | − 1.49·3-s + 1.23·9-s + 1.41·11-s + 0.717·13-s + 0.440·17-s + 1.36·19-s − 1.54·23-s − 0.349·27-s − 1.24·29-s − 0.277·31-s − 2.11·33-s − 1.21·37-s − 1.07·39-s + 0.241·41-s − 0.609·43-s + 1.54·47-s − 0.658·51-s − 1.56·53-s − 2.04·57-s − 1.72·59-s + 0.366·61-s − 0.977·67-s + 2.30·69-s + 0.878·71-s − 1.45·73-s − 0.911·79-s − 0.711·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 - 1.54T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 - 2.85T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 8.53T + 97T^{2} \) |
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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
−7.22641807684434322942658486605, −6.43355506893543705587252047395, −5.93959479188296081057633413294, −5.50575548717016983236106656827, −4.64861364549522201860331008376, −3.89008540063509879921097488346, −3.29492606294194192277514378393, −1.74318184197052990340860898513, −1.15882144866024174080270011825, 0,
1.15882144866024174080270011825, 1.74318184197052990340860898513, 3.29492606294194192277514378393, 3.89008540063509879921097488346, 4.64861364549522201860331008376, 5.50575548717016983236106656827, 5.93959479188296081057633413294, 6.43355506893543705587252047395, 7.22641807684434322942658486605
