| L(s) = 1 | + 2.56·3-s − 5-s − 0.561·7-s + 3.56·9-s + 11-s + 5.12·13-s − 2.56·15-s + 1.43·17-s + 6.56·19-s − 1.43·21-s − 1.12·23-s + 25-s + 1.43·27-s − 4.56·29-s − 3.68·31-s + 2.56·33-s + 0.561·35-s − 10.8·37-s + 13.1·39-s − 10·41-s − 3.12·43-s − 3.56·45-s + 1.12·47-s − 6.68·49-s + 3.68·51-s − 8.56·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.447·5-s − 0.212·7-s + 1.18·9-s + 0.301·11-s + 1.42·13-s − 0.661·15-s + 0.348·17-s + 1.50·19-s − 0.313·21-s − 0.234·23-s + 0.200·25-s + 0.276·27-s − 0.847·29-s − 0.661·31-s + 0.445·33-s + 0.0949·35-s − 1.77·37-s + 2.10·39-s − 1.56·41-s − 0.476·43-s − 0.530·45-s + 0.163·47-s − 0.954·49-s + 0.515·51-s − 1.17·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143814635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143814635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.561T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 8.87T + 97T^{2} \) |
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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
−11.13591498300552071401281724256, −9.976028038417769845935815433173, −9.154380396298334432200067721256, −8.436898456575718341715898561019, −7.68770624220642610138253092199, −6.71191935532319040866435937646, −5.31085857623182783319895761315, −3.65036637553088160903871492399, −3.36249908451331144989050728306, −1.65544387779619858369708050137,
1.65544387779619858369708050137, 3.36249908451331144989050728306, 3.65036637553088160903871492399, 5.31085857623182783319895761315, 6.71191935532319040866435937646, 7.68770624220642610138253092199, 8.436898456575718341715898561019, 9.154380396298334432200067721256, 9.976028038417769845935815433173, 11.13591498300552071401281724256
