L(s) = 1 | − 3-s − 0.547·7-s + 9-s − 1.33·11-s − 0.302·13-s − 4.04·17-s − 5.63·19-s + 0.547·21-s − 0.245·23-s − 27-s − 1.36·29-s − 3.53·31-s + 1.33·33-s + 2.18·37-s + 0.302·39-s − 9.80·41-s + 8.35·43-s + 10.4·47-s − 6.70·49-s + 4.04·51-s + 7.69·53-s + 5.63·57-s + 4.15·59-s + 2.07·61-s − 0.547·63-s − 8.48·67-s + 0.245·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.206·7-s + 0.333·9-s − 0.403·11-s − 0.0837·13-s − 0.980·17-s − 1.29·19-s + 0.119·21-s − 0.0511·23-s − 0.192·27-s − 0.254·29-s − 0.635·31-s + 0.232·33-s + 0.359·37-s + 0.0483·39-s − 1.53·41-s + 1.27·43-s + 1.52·47-s − 0.957·49-s + 0.565·51-s + 1.05·53-s + 0.746·57-s + 0.540·59-s + 0.265·61-s − 0.0689·63-s − 1.03·67-s + 0.0295·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9205107589\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9205107589\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.547T + 7T^{2} \) |
| 11 | \( 1 + 1.33T + 11T^{2} \) |
| 13 | \( 1 + 0.302T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 23 | \( 1 + 0.245T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 - 8.35T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 - 2.07T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 - 4.97T + 89T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−7.74270698563308535331780107004, −7.16184799677640642342846080823, −6.40888128262342069672106076050, −5.90348369664243957561732152019, −5.06404917216838292080614759272, −4.38616272023069216675257502412, −3.68935844665057733038193319011, −2.56516334850384817411971906076, −1.84407812410726899417911309507, −0.47788715665751704372982538522,
0.47788715665751704372982538522, 1.84407812410726899417911309507, 2.56516334850384817411971906076, 3.68935844665057733038193319011, 4.38616272023069216675257502412, 5.06404917216838292080614759272, 5.90348369664243957561732152019, 6.40888128262342069672106076050, 7.16184799677640642342846080823, 7.74270698563308535331780107004