| L(s) = 1 | + 0.538·3-s + 0.810·5-s − 2.70·9-s + 4.92·11-s + 0.0654·13-s + 0.436·15-s − 1.24·17-s + 3.82·19-s − 1.71·23-s − 4.34·25-s − 3.07·27-s + 3.89·29-s + 3.19·31-s + 2.65·33-s − 0.0930·37-s + 0.0352·39-s − 41-s + 10.5·43-s − 2.19·45-s − 0.959·47-s − 0.668·51-s − 2.72·53-s + 3.99·55-s + 2.06·57-s + 11.1·59-s − 10.8·61-s + 0.0530·65-s + ⋯ |
| L(s) = 1 | + 0.311·3-s + 0.362·5-s − 0.903·9-s + 1.48·11-s + 0.0181·13-s + 0.112·15-s − 0.300·17-s + 0.877·19-s − 0.358·23-s − 0.868·25-s − 0.592·27-s + 0.724·29-s + 0.574·31-s + 0.462·33-s − 0.0152·37-s + 0.00564·39-s − 0.156·41-s + 1.61·43-s − 0.327·45-s − 0.139·47-s − 0.0935·51-s − 0.374·53-s + 0.538·55-s + 0.272·57-s + 1.45·59-s − 1.38·61-s + 0.00657·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534744122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534744122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.538T + 3T^{2} \) |
| 5 | \( 1 - 0.810T + 5T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 - 0.0654T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 0.0930T + 37T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.959T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 6.31T + 97T^{2} \) |
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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
−7.87358120838837014386997985431, −7.15905952752413323590543393472, −6.27381204636410871669909957608, −5.96318255424222691908383277801, −5.05748958728778189380939293233, −4.17597334799446651045481306338, −3.50474589688787021113056034616, −2.67373014487682916019203955527, −1.82194441739813767944674370911, −0.78884935137199995048637517826,
0.78884935137199995048637517826, 1.82194441739813767944674370911, 2.67373014487682916019203955527, 3.50474589688787021113056034616, 4.17597334799446651045481306338, 5.05748958728778189380939293233, 5.96318255424222691908383277801, 6.27381204636410871669909957608, 7.15905952752413323590543393472, 7.87358120838837014386997985431
