L(s) = 1 | + 3-s + 3·7-s + 9-s − 2·11-s + 13-s + 2·17-s + 5·19-s + 3·21-s − 6·23-s + 27-s + 10·29-s + 3·31-s − 2·33-s + 2·37-s + 39-s − 8·41-s − 43-s − 2·47-s + 2·49-s + 2·51-s − 4·53-s + 5·57-s + 10·59-s + 7·61-s + 3·63-s + 3·67-s − 6·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s + 0.192·27-s + 1.85·29-s + 0.538·31-s − 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s + 0.280·51-s − 0.549·53-s + 0.662·57-s + 1.30·59-s + 0.896·61-s + 0.377·63-s + 0.366·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.334766374\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.334766374\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.946237306464798320060676890295, −8.686228257120952969207606021954, −8.134034752267247325522916900531, −7.57404393602003080106109524301, −6.47388999507393793454845833318, −5.34490029281998815763412767656, −4.63718444572594903228967418102, −3.50818980571584578672789924466, −2.44041792194616260799140408514, −1.24790523142424415923064013417,
1.24790523142424415923064013417, 2.44041792194616260799140408514, 3.50818980571584578672789924466, 4.63718444572594903228967418102, 5.34490029281998815763412767656, 6.47388999507393793454845833318, 7.57404393602003080106109524301, 8.134034752267247325522916900531, 8.686228257120952969207606021954, 9.946237306464798320060676890295