| L(s) = 1 | + 0.347·3-s − 3.41·7-s − 2.87·9-s − 2.41·11-s − 6.29·13-s + 2.34·17-s − 19-s − 1.18·21-s + 2.49·23-s − 2.04·27-s − 8.17·29-s + 2.77·31-s − 0.837·33-s + 0.977·37-s − 2.18·39-s − 3.49·41-s − 2.75·43-s + 6.29·47-s + 4.63·49-s + 0.815·51-s − 2.38·53-s − 0.347·57-s − 3.67·59-s − 12.7·61-s + 9.82·63-s + 2.41·67-s + 0.864·69-s + ⋯ |
| L(s) = 1 | + 0.200·3-s − 1.28·7-s − 0.959·9-s − 0.727·11-s − 1.74·13-s + 0.569·17-s − 0.229·19-s − 0.258·21-s + 0.519·23-s − 0.392·27-s − 1.51·29-s + 0.498·31-s − 0.145·33-s + 0.160·37-s − 0.349·39-s − 0.545·41-s − 0.420·43-s + 0.917·47-s + 0.662·49-s + 0.114·51-s − 0.328·53-s − 0.0460·57-s − 0.478·59-s − 1.63·61-s + 1.23·63-s + 0.294·67-s + 0.104·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5342756027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5342756027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 - 0.977T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 - 5.04T + 79T^{2} \) |
| 83 | \( 1 + 8.07T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−7.74262618998870008336593220384, −7.29544724653658116554568150091, −6.50586456247509636999060616743, −5.71566838890555972277452682666, −5.21990688596653368404607764642, −4.31451572698806447668483988503, −3.18023750315256757473232203969, −2.91532897800287268285990493579, −2.01951643583763247980733042587, −0.33242022739631624805323063629,
0.33242022739631624805323063629, 2.01951643583763247980733042587, 2.91532897800287268285990493579, 3.18023750315256757473232203969, 4.31451572698806447668483988503, 5.21990688596653368404607764642, 5.71566838890555972277452682666, 6.50586456247509636999060616743, 7.29544724653658116554568150091, 7.74262618998870008336593220384
