| L(s) = 1 | − 2-s + 2.04·3-s + 4-s + 2.35·5-s − 2.04·6-s + 0.890·7-s − 8-s + 1.19·9-s − 2.35·10-s + 2.49·11-s + 2.04·12-s − 0.890·14-s + 4.82·15-s + 16-s + 3.58·17-s − 1.19·18-s + 19-s + 2.35·20-s + 1.82·21-s − 2.49·22-s + 3.20·23-s − 2.04·24-s + 0.554·25-s − 3.69·27-s + 0.890·28-s − 0.219·29-s − 4.82·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.05·5-s − 0.836·6-s + 0.336·7-s − 0.353·8-s + 0.399·9-s − 0.745·10-s + 0.751·11-s + 0.591·12-s − 0.237·14-s + 1.24·15-s + 0.250·16-s + 0.868·17-s − 0.282·18-s + 0.229·19-s + 0.527·20-s + 0.397·21-s − 0.531·22-s + 0.668·23-s − 0.418·24-s + 0.110·25-s − 0.710·27-s + 0.168·28-s − 0.0408·29-s − 0.881·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.280511218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.280511218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 2.35T + 5T^{2} \) |
| 7 | \( 1 - 0.890T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 + 0.219T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 + 0.0978T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 + 3.34T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 0.121T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−8.100001911634114351056214405500, −7.60854521282295325319715968774, −6.71619311155863946537201578647, −6.04576906638281806548857114263, −5.31066361899982050410312857264, −4.26150990322836600933224721972, −3.27212711016406323918686664182, −2.64614360175999979129435071590, −1.81465310702634958167676887713, −1.06646329791061256447023831864,
1.06646329791061256447023831864, 1.81465310702634958167676887713, 2.64614360175999979129435071590, 3.27212711016406323918686664182, 4.26150990322836600933224721972, 5.31066361899982050410312857264, 6.04576906638281806548857114263, 6.71619311155863946537201578647, 7.60854521282295325319715968774, 8.100001911634114351056214405500
