| L(s) = 1 | + 1.34·3-s − 3.42·5-s − 1.69·7-s − 1.19·9-s − 3.25·11-s − 2.67·13-s − 4.59·15-s + 2·17-s + 6.95·19-s − 2.27·21-s + 7.30·23-s + 6.72·25-s − 5.63·27-s − 2.67·29-s − 2.70·31-s − 4.37·33-s + 5.80·35-s − 37-s − 3.58·39-s + 8.94·41-s + 7.66·43-s + 4.09·45-s + 7.50·47-s − 4.12·49-s + 2.68·51-s + 8.66·53-s + 11.1·55-s + ⋯ |
| L(s) = 1 | + 0.775·3-s − 1.53·5-s − 0.640·7-s − 0.398·9-s − 0.981·11-s − 0.740·13-s − 1.18·15-s + 0.485·17-s + 1.59·19-s − 0.496·21-s + 1.52·23-s + 1.34·25-s − 1.08·27-s − 0.496·29-s − 0.486·31-s − 0.761·33-s + 0.981·35-s − 0.164·37-s − 0.574·39-s + 1.39·41-s + 1.16·43-s + 0.610·45-s + 1.09·47-s − 0.589·49-s + 0.376·51-s + 1.19·53-s + 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172262100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172262100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 + 2.70T + 31T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 + 0.576T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 7.30T + 79T^{2} \) |
| 83 | \( 1 + 0.990T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.095928626658449208538444277895, −7.949003799498513707122672059699, −7.61518114660043356625493383028, −7.10492898956395748944348042667, −5.69949606122437804184567756105, −4.98084579356673678649618921249, −3.86511359359805130047185739695, −3.15568833185674849763441464527, −2.61425801610316189603757033436, −0.64182076350127137841512975024,
0.64182076350127137841512975024, 2.61425801610316189603757033436, 3.15568833185674849763441464527, 3.86511359359805130047185739695, 4.98084579356673678649618921249, 5.69949606122437804184567756105, 7.10492898956395748944348042667, 7.61518114660043356625493383028, 7.949003799498513707122672059699, 9.095928626658449208538444277895
