L(s) = 1 | + 2-s − 3.23·3-s + 4-s + 5-s − 3.23·6-s + 4.13·7-s + 8-s + 7.46·9-s + 10-s + 3.69·11-s − 3.23·12-s + 4.13·14-s − 3.23·15-s + 16-s + 4.39·17-s + 7.46·18-s − 4.02·19-s + 20-s − 13.3·21-s + 3.69·22-s − 4.90·23-s − 3.23·24-s + 25-s − 14.4·27-s + 4.13·28-s + 5.10·29-s − 3.23·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s + 0.447·5-s − 1.32·6-s + 1.56·7-s + 0.353·8-s + 2.48·9-s + 0.316·10-s + 1.11·11-s − 0.933·12-s + 1.10·14-s − 0.835·15-s + 0.250·16-s + 1.06·17-s + 1.76·18-s − 0.924·19-s + 0.223·20-s − 2.92·21-s + 0.788·22-s − 1.02·23-s − 0.660·24-s + 0.200·25-s − 2.78·27-s + 0.782·28-s + 0.947·29-s − 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.136410462\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.136410462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 5.10T + 29T^{2} \) |
| 31 | \( 1 - 0.517T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 + 0.988T + 41T^{2} \) |
| 43 | \( 1 - 7.45T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 - 4.96T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 - 0.530T + 67T^{2} \) |
| 71 | \( 1 - 0.121T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 4.40T + 97T^{2} \) |
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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
−9.691649808731794874415715280293, −8.379574607181880679382111416218, −7.47655704912160963781344957669, −6.53674774750217811045209284306, −6.02934517071443307988073950500, −5.22444395351060863676775770160, −4.64438551416181032074464120841, −3.89171007853053902353575587265, −1.91863179775361326241207035816, −1.12522637205893208122649263847,
1.12522637205893208122649263847, 1.91863179775361326241207035816, 3.89171007853053902353575587265, 4.64438551416181032074464120841, 5.22444395351060863676775770160, 6.02934517071443307988073950500, 6.53674774750217811045209284306, 7.47655704912160963781344957669, 8.379574607181880679382111416218, 9.691649808731794874415715280293