| L(s) = 1 | − 3-s + 5-s − 2.58·7-s + 9-s − 0.861·11-s + 3.72·13-s − 15-s + 5.01·17-s − 2.31·19-s + 2.58·21-s − 5.33·23-s + 25-s − 27-s + 6.19·29-s − 31-s + 0.861·33-s − 2.58·35-s + 1.72·37-s − 3.72·39-s − 5.48·41-s + 3.77·43-s + 45-s − 0.437·47-s − 0.299·49-s − 5.01·51-s − 8.82·53-s − 0.861·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.978·7-s + 0.333·9-s − 0.259·11-s + 1.03·13-s − 0.258·15-s + 1.21·17-s − 0.531·19-s + 0.564·21-s − 1.11·23-s + 0.200·25-s − 0.192·27-s + 1.15·29-s − 0.179·31-s + 0.149·33-s − 0.437·35-s + 0.283·37-s − 0.596·39-s − 0.857·41-s + 0.574·43-s + 0.149·45-s − 0.0638·47-s − 0.0427·49-s − 0.702·51-s − 1.21·53-s − 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419548937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419548937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 + 0.861T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 37 | \( 1 - 1.72T + 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 + 0.437T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 + 4.90T + 61T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.406924377448545225415057137830, −7.87977747919328268660990348147, −6.73451382336682115973757016280, −6.28355279020358978159715726221, −5.70688938886268963615589982102, −4.84515298845472613607460249866, −3.79954432989123038115383474211, −3.10756776765516579937375718606, −1.90471728286255752725907180518, −0.71690769440226925882490940743,
0.71690769440226925882490940743, 1.90471728286255752725907180518, 3.10756776765516579937375718606, 3.79954432989123038115383474211, 4.84515298845472613607460249866, 5.70688938886268963615589982102, 6.28355279020358978159715726221, 6.73451382336682115973757016280, 7.87977747919328268660990348147, 8.406924377448545225415057137830
