| L(s) = 1 | − 1.56·3-s + 1.32·5-s − 4.71·7-s − 0.561·9-s − 1.32·11-s + 0.438·13-s − 2.06·15-s − 4.71·17-s − 6.04·19-s + 7.36·21-s − 3.24·25-s + 5.56·27-s − 0.438·29-s − 5.56·31-s + 2.06·33-s − 6.24·35-s − 8.10·37-s − 0.684·39-s + 10.6·41-s − 8.68·43-s − 0.743·45-s − 11.8·47-s + 15.2·49-s + 7.36·51-s − 6.04·53-s − 1.75·55-s + 9.43·57-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.592·5-s − 1.78·7-s − 0.187·9-s − 0.399·11-s + 0.121·13-s − 0.533·15-s − 1.14·17-s − 1.38·19-s + 1.60·21-s − 0.649·25-s + 1.07·27-s − 0.0814·29-s − 0.998·31-s + 0.359·33-s − 1.05·35-s − 1.33·37-s − 0.109·39-s + 1.66·41-s − 1.32·43-s − 0.110·45-s − 1.72·47-s + 2.17·49-s + 1.03·51-s − 0.829·53-s − 0.236·55-s + 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05749125103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05749125103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.64636397595182623957159008918, −6.60206276263202963029259670920, −6.50247330885344595396466667186, −5.87336511198865914994402615679, −5.20452956638053746126795267677, −4.31794644808171434221425497413, −3.44598564779092151422718303354, −2.66276787494919504738115542619, −1.79378174324942933228922635109, −0.11450491978056290294721151479,
0.11450491978056290294721151479, 1.79378174324942933228922635109, 2.66276787494919504738115542619, 3.44598564779092151422718303354, 4.31794644808171434221425497413, 5.20452956638053746126795267677, 5.87336511198865914994402615679, 6.50247330885344595396466667186, 6.60206276263202963029259670920, 7.64636397595182623957159008918
