L(s) = 1 | − 3-s + 5-s − 2.82·7-s + 9-s − 4·11-s − 13-s − 15-s + 4.82·17-s + 2.82·19-s + 2.82·21-s − 2.82·23-s + 25-s − 27-s − 4.82·29-s + 9.65·31-s + 4·33-s − 2.82·35-s + 7.65·37-s + 39-s + 0.343·41-s − 9.65·43-s + 45-s + 1.65·47-s + 1.00·49-s − 4.82·51-s + 13.3·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.06·7-s + 0.333·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 1.17·17-s + 0.648·19-s + 0.617·21-s − 0.589·23-s + 0.200·25-s − 0.192·27-s − 0.896·29-s + 1.73·31-s + 0.696·33-s − 0.478·35-s + 1.25·37-s + 0.160·39-s + 0.0535·41-s − 1.47·43-s + 0.149·45-s + 0.241·47-s + 0.142·49-s − 0.676·51-s + 1.82·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.69857981961503792669088667116, −6.88815625544501699846327504972, −6.19859125501433445460824016676, −5.56534966430287245836261513492, −5.06839996635424300269417879593, −4.02942235834725422079721166916, −3.09048257757754632206312473517, −2.46412081208854751346991094724, −1.14360649981184563108810772519, 0,
1.14360649981184563108810772519, 2.46412081208854751346991094724, 3.09048257757754632206312473517, 4.02942235834725422079721166916, 5.06839996635424300269417879593, 5.56534966430287245836261513492, 6.19859125501433445460824016676, 6.88815625544501699846327504972, 7.69857981961503792669088667116