| L(s) = 1 | + 1.26·3-s − 11.0·5-s − 5.70·7-s − 25.4·9-s + 59.0·11-s − 88.0·13-s − 13.9·15-s − 113.·17-s + 19·19-s − 7.20·21-s − 40.1·23-s − 3.28·25-s − 66.1·27-s + 66.5·29-s + 248.·31-s + 74.5·33-s + 62.9·35-s + 330.·37-s − 111.·39-s − 172.·41-s + 56.9·43-s + 280.·45-s − 483.·47-s − 310.·49-s − 142.·51-s + 104.·53-s − 651.·55-s + ⋯ |
| L(s) = 1 | + 0.243·3-s − 0.986·5-s − 0.308·7-s − 0.940·9-s + 1.61·11-s − 1.87·13-s − 0.239·15-s − 1.61·17-s + 0.229·19-s − 0.0748·21-s − 0.364·23-s − 0.0263·25-s − 0.471·27-s + 0.426·29-s + 1.44·31-s + 0.393·33-s + 0.303·35-s + 1.47·37-s − 0.456·39-s − 0.655·41-s + 0.201·43-s + 0.928·45-s − 1.49·47-s − 0.905·49-s − 0.392·51-s + 0.270·53-s − 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9688243824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9688243824\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 1.26T + 27T^{2} \) |
| 5 | \( 1 + 11.0T + 125T^{2} \) |
| 7 | \( 1 + 5.70T + 343T^{2} \) |
| 11 | \( 1 - 59.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 40.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 66.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 56.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 579.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 314.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 12.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 711.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 704.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 50.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 849.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 704.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 232.T + 9.12e5T^{2} \) |
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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
−9.350690449385044152810234523254, −8.509497778090091789344524342323, −7.81461553108765711692471587597, −6.83596105450394847408695626949, −6.26513972749271582466249441320, −4.83247395786834814031413711362, −4.18998373505915427460339736425, −3.14688313383011954947005983290, −2.18620697695008997151843358445, −0.47198517457132563180588424559,
0.47198517457132563180588424559, 2.18620697695008997151843358445, 3.14688313383011954947005983290, 4.18998373505915427460339736425, 4.83247395786834814031413711362, 6.26513972749271582466249441320, 6.83596105450394847408695626949, 7.81461553108765711692471587597, 8.509497778090091789344524342323, 9.350690449385044152810234523254
