L(s) = 1 | + 3.44·3-s − 9.69·5-s − 27.5·7-s − 15.1·9-s + 52.3·11-s + 5.40·13-s − 33.4·15-s + 17.1·17-s − 44.2·19-s − 95.1·21-s − 205.·23-s − 30.9·25-s − 145.·27-s − 29·29-s + 299.·31-s + 180.·33-s + 267.·35-s − 29.7·37-s + 18.6·39-s − 43.9·41-s − 64.8·43-s + 146.·45-s − 499.·47-s + 418.·49-s + 59.3·51-s + 351.·53-s − 507.·55-s + ⋯ |
L(s) = 1 | + 0.663·3-s − 0.867·5-s − 1.49·7-s − 0.559·9-s + 1.43·11-s + 0.115·13-s − 0.575·15-s + 0.245·17-s − 0.533·19-s − 0.989·21-s − 1.85·23-s − 0.247·25-s − 1.03·27-s − 0.185·29-s + 1.73·31-s + 0.952·33-s + 1.29·35-s − 0.132·37-s + 0.0765·39-s − 0.167·41-s − 0.229·43-s + 0.485·45-s − 1.55·47-s + 1.22·49-s + 0.162·51-s + 0.911·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.154925839\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.154925839\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
good | 3 | \( 1 - 3.44T + 27T^{2} \) |
| 5 | \( 1 + 9.69T + 125T^{2} \) |
| 7 | \( 1 + 27.5T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.40T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 29.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 351.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 504.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 481.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.11T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 295.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 428.T + 9.12e5T^{2} \) |
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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
−8.769479737751325647048835913566, −8.244756372905439831809539320694, −7.39741431500271757074391105961, −6.31928024839069932214993420526, −6.10804389688195153774177041288, −4.45592025332053249081284439596, −3.65226383290654908420890471325, −3.22588952422120851375556921165, −2.01156523622587117318376398948, −0.46397687471829912936621347208,
0.46397687471829912936621347208, 2.01156523622587117318376398948, 3.22588952422120851375556921165, 3.65226383290654908420890471325, 4.45592025332053249081284439596, 6.10804389688195153774177041288, 6.31928024839069932214993420526, 7.39741431500271757074391105961, 8.244756372905439831809539320694, 8.769479737751325647048835913566