L(s) = 1 | + 3-s − 5-s + 2·7-s − 2·9-s + 4·13-s − 15-s + 2·17-s + 2·21-s − 23-s − 4·25-s − 5·27-s + 7·31-s − 2·35-s − 3·37-s + 4·39-s + 8·41-s − 6·43-s + 2·45-s + 8·47-s − 3·49-s + 2·51-s + 6·53-s − 5·59-s + 12·61-s − 4·63-s − 4·65-s + 7·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.436·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.25·31-s − 0.338·35-s − 0.493·37-s + 0.640·39-s + 1.24·41-s − 0.914·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.650·59-s + 1.53·61-s − 0.503·63-s − 0.496·65-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.488165982\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.488165982\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.986977214961441919258861299065, −7.43339040852683851690755796771, −6.41739922474452313355276923719, −5.78810185028013809349253578353, −5.06178318871990400703439654803, −4.10720691214350831565330314138, −3.59045856507361411076663144986, −2.72065307859336189385477919217, −1.84744665516122104676675056785, −0.77578045286233029832552663292,
0.77578045286233029832552663292, 1.84744665516122104676675056785, 2.72065307859336189385477919217, 3.59045856507361411076663144986, 4.10720691214350831565330314138, 5.06178318871990400703439654803, 5.78810185028013809349253578353, 6.41739922474452313355276923719, 7.43339040852683851690755796771, 7.986977214961441919258861299065