| L(s) = 1 | + 3·3-s − 5·5-s − 5.14·7-s + 9·9-s + 65.5·11-s + 10.5·13-s − 15·15-s + 99.9·17-s − 31.9·19-s − 15.4·21-s − 23·23-s + 25·25-s + 27·27-s + 94.8·29-s − 339.·31-s + 196.·33-s + 25.7·35-s + 183.·37-s + 31.7·39-s + 158.·41-s − 41.6·43-s − 45·45-s − 457.·47-s − 316.·49-s + 299.·51-s − 154.·53-s − 327.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.277·7-s + 0.333·9-s + 1.79·11-s + 0.225·13-s − 0.258·15-s + 1.42·17-s − 0.385·19-s − 0.160·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.607·29-s − 1.96·31-s + 1.03·33-s + 0.124·35-s + 0.813·37-s + 0.130·39-s + 0.604·41-s − 0.147·43-s − 0.149·45-s − 1.42·47-s − 0.922·49-s + 0.823·51-s − 0.399·53-s − 0.803·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.847445615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.847445615\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 7 | \( 1 + 5.14T + 343T^{2} \) |
| 11 | \( 1 - 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 94.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 339.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 183.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 158.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 154.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 368.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 824.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 801.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 818.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 46.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 398.T + 9.12e5T^{2} \) |
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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
−9.356595677461218880158674030020, −8.370821879983088600082936298769, −7.71005045379123811881143303374, −6.76938164720489006076247478629, −6.09887619718980169186649130438, −4.86336893791639852841336550618, −3.74433055992601679637455839183, −3.40509117454194919418522675269, −1.90150799514146004162820622455, −0.855426331963787890859257573506,
0.855426331963787890859257573506, 1.90150799514146004162820622455, 3.40509117454194919418522675269, 3.74433055992601679637455839183, 4.86336893791639852841336550618, 6.09887619718980169186649130438, 6.76938164720489006076247478629, 7.71005045379123811881143303374, 8.370821879983088600082936298769, 9.356595677461218880158674030020
