| L(s) = 1 | − 3-s − 2·9-s − 5·11-s − 5·13-s − 17-s − 6·19-s + 4·23-s − 5·25-s + 5·27-s + 4·29-s + 5·33-s + 8·37-s + 5·39-s + 4·41-s − 6·43-s − 6·47-s + 51-s + 11·53-s + 6·57-s + 10·59-s + 10·67-s − 4·69-s + 9·71-s + 4·73-s + 5·75-s + 9·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.50·11-s − 1.38·13-s − 0.242·17-s − 1.37·19-s + 0.834·23-s − 25-s + 0.962·27-s + 0.742·29-s + 0.870·33-s + 1.31·37-s + 0.800·39-s + 0.624·41-s − 0.914·43-s − 0.875·47-s + 0.140·51-s + 1.51·53-s + 0.794·57-s + 1.30·59-s + 1.22·67-s − 0.481·69-s + 1.06·71-s + 0.468·73-s + 0.577·75-s + 1.01·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6403012744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6403012744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.374012440735736479868877176293, −8.042300812665714533613275999044, −7.04860562711275482218570288402, −6.38511506014092420612219539214, −5.40864521517896805635688508757, −5.04360908944665888730354415016, −4.11321335610142599569466903869, −2.73780970066283753303466245103, −2.31208754219170212371373460686, −0.45582208180966446071572557340,
0.45582208180966446071572557340, 2.31208754219170212371373460686, 2.73780970066283753303466245103, 4.11321335610142599569466903869, 5.04360908944665888730354415016, 5.40864521517896805635688508757, 6.38511506014092420612219539214, 7.04860562711275482218570288402, 8.042300812665714533613275999044, 8.374012440735736479868877176293
