| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 3.41·11-s + 1.17·13-s + 16-s + 3.41·17-s + 4.82·19-s − 20-s − 3.41·22-s + 3.65·23-s + 25-s + 1.17·26-s − 5.41·29-s + 7.41·31-s + 32-s + 3.41·34-s − 3.41·37-s + 4.82·38-s − 40-s − 3.65·41-s − 7.07·43-s − 3.41·44-s + 3.65·46-s + 2.58·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.02·11-s + 0.324·13-s + 0.250·16-s + 0.828·17-s + 1.10·19-s − 0.223·20-s − 0.727·22-s + 0.762·23-s + 0.200·25-s + 0.229·26-s − 1.00·29-s + 1.33·31-s + 0.176·32-s + 0.585·34-s − 0.561·37-s + 0.783·38-s − 0.158·40-s − 0.571·41-s − 1.07·43-s − 0.514·44-s + 0.539·46-s + 0.377·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808136947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808136947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 0.828T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 0.485T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.040264317826222713448324677190, −7.70046186847754064570118579352, −6.87595568515737267107206846034, −6.08431808317092472635219522169, −5.12704258308494886166040579866, −4.90931216342004858491689244602, −3.56138586346781589082217564499, −3.24010507312999346875706925640, −2.13105282539703761973567226224, −0.853874032457850781213992552000,
0.853874032457850781213992552000, 2.13105282539703761973567226224, 3.24010507312999346875706925640, 3.56138586346781589082217564499, 4.90931216342004858491689244602, 5.12704258308494886166040579866, 6.08431808317092472635219522169, 6.87595568515737267107206846034, 7.70046186847754064570118579352, 8.040264317826222713448324677190
