| L(s) = 1 | + 2.79·3-s − 3·5-s + 4.79·9-s − 3.79·11-s + 13-s − 8.37·15-s − 3.79·17-s − 19-s + 4.58·23-s + 4·25-s + 4.99·27-s + 3.79·29-s − 7.37·31-s − 10.5·33-s + 5·37-s + 2.79·39-s + 3.79·41-s + 2·43-s − 14.3·45-s − 10.5·47-s − 10.5·51-s − 8.37·53-s + 11.3·55-s − 2.79·57-s − 12.1·59-s + 61-s − 3·65-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 1.34·5-s + 1.59·9-s − 1.14·11-s + 0.277·13-s − 2.16·15-s − 0.919·17-s − 0.229·19-s + 0.955·23-s + 0.800·25-s + 0.962·27-s + 0.704·29-s − 1.32·31-s − 1.84·33-s + 0.821·37-s + 0.446·39-s + 0.592·41-s + 0.304·43-s − 2.14·45-s − 1.54·47-s − 1.48·51-s − 1.15·53-s + 1.53·55-s − 0.369·57-s − 1.58·59-s + 0.128·61-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 8.37T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.216183052598718612845774813413, −7.50852595263437380552743657165, −7.19552896582901802086673431869, −5.99061221415617308666379916393, −4.66033348580179924303350891340, −4.26575391360940547414690762090, −3.17371881328271977116196449846, −2.87538700715343684786860866502, −1.68144048209002389236322480362, 0,
1.68144048209002389236322480362, 2.87538700715343684786860866502, 3.17371881328271977116196449846, 4.26575391360940547414690762090, 4.66033348580179924303350891340, 5.99061221415617308666379916393, 7.19552896582901802086673431869, 7.50852595263437380552743657165, 8.216183052598718612845774813413
