| L(s) = 1 | + 1.08·2-s − 0.824·4-s − 0.786·5-s − 4.63·7-s − 3.06·8-s − 0.852·10-s − 4.80·11-s − 3.44·13-s − 5.02·14-s − 1.67·16-s + 1.46·17-s − 1.81·19-s + 0.648·20-s − 5.20·22-s + 23-s − 4.38·25-s − 3.73·26-s + 3.82·28-s + 29-s − 4.86·31-s + 4.31·32-s + 1.58·34-s + 3.64·35-s − 10.3·37-s − 1.96·38-s + 2.40·40-s − 9.91·41-s + ⋯ |
| L(s) = 1 | + 0.766·2-s − 0.412·4-s − 0.351·5-s − 1.75·7-s − 1.08·8-s − 0.269·10-s − 1.44·11-s − 0.955·13-s − 1.34·14-s − 0.417·16-s + 0.354·17-s − 0.415·19-s + 0.144·20-s − 1.11·22-s + 0.208·23-s − 0.876·25-s − 0.732·26-s + 0.722·28-s + 0.185·29-s − 0.874·31-s + 0.762·32-s + 0.271·34-s + 0.615·35-s − 1.70·37-s − 0.318·38-s + 0.380·40-s − 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08730601636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08730601636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 5 | \( 1 + 0.786T + 5T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 + 6.59T + 43T^{2} \) |
| 47 | \( 1 + 2.49T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 - 7.02T + 71T^{2} \) |
| 73 | \( 1 + 6.41T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−8.129946938608816677100645531757, −7.12073026480471023601837977649, −6.70459301767302028087718791453, −5.62332121840361408286693728465, −5.34003541054462826575382406153, −4.43842578380233438063771196639, −3.48147935784870602888170300331, −3.16785523791951196847242384707, −2.22678440759361976837872088962, −0.12634662191328711405444781632,
0.12634662191328711405444781632, 2.22678440759361976837872088962, 3.16785523791951196847242384707, 3.48147935784870602888170300331, 4.43842578380233438063771196639, 5.34003541054462826575382406153, 5.62332121840361408286693728465, 6.70459301767302028087718791453, 7.12073026480471023601837977649, 8.129946938608816677100645531757
