L(s) = 1 | − 3-s + 3.73·5-s + 2.73·7-s + 9-s − 1.26·11-s − 3.73·15-s − 5.73·17-s − 4.73·19-s − 2.73·21-s − 4.19·23-s + 8.92·25-s − 27-s − 4.46·29-s − 1.46·31-s + 1.26·33-s + 10.1·35-s + 3.53·37-s − 9.39·41-s + 9.66·43-s + 3.73·45-s − 2.19·47-s + 0.464·49-s + 5.73·51-s − 6.46·53-s − 4.73·55-s + 4.73·57-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.66·5-s + 1.03·7-s + 0.333·9-s − 0.382·11-s − 0.963·15-s − 1.39·17-s − 1.08·19-s − 0.596·21-s − 0.874·23-s + 1.78·25-s − 0.192·27-s − 0.828·29-s − 0.262·31-s + 0.220·33-s + 1.72·35-s + 0.581·37-s − 1.46·41-s + 1.47·43-s + 0.556·45-s − 0.320·47-s + 0.0663·49-s + 0.802·51-s − 0.887·53-s − 0.638·55-s + 0.626·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 - 9.66T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 9.19T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42236257052305934378912313068, −6.52766643304503246522182889177, −6.07236155478885992200702038540, −5.49672643110538817039225471177, −4.71221874543091718902656302996, −4.26787147449301197560380364074, −2.82109530102471290168517646616, −1.90859250755270574760087188778, −1.63232394227075698580757337514, 0,
1.63232394227075698580757337514, 1.90859250755270574760087188778, 2.82109530102471290168517646616, 4.26787147449301197560380364074, 4.71221874543091718902656302996, 5.49672643110538817039225471177, 6.07236155478885992200702038540, 6.52766643304503246522182889177, 7.42236257052305934378912313068