| L(s) = 1 | + 4.46·5-s + 0.464·13-s − 7.92·17-s + 14.9·25-s + 8.46·29-s + 11.3·37-s + 10·41-s − 7·49-s + 14·53-s − 5.39·61-s + 2.07·65-s − 10.8·73-s − 35.3·85-s + 8.85·89-s + 18·97-s − 2·101-s − 20.3·109-s − 6.85·113-s + ⋯ |
| L(s) = 1 | + 1.99·5-s + 0.128·13-s − 1.92·17-s + 2.98·25-s + 1.57·29-s + 1.87·37-s + 1.56·41-s − 49-s + 1.92·53-s − 0.690·61-s + 0.256·65-s − 1.27·73-s − 3.83·85-s + 0.938·89-s + 1.82·97-s − 0.199·101-s − 1.94·109-s − 0.644·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.711440905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711440905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.076439298074847786705740817908, −8.346745577319619020387011706250, −7.14175823931883657034929116512, −6.31968028107713771021307671812, −6.03847146684613022794811719368, −4.98408235175799821748161370558, −4.31834572542020013709828773845, −2.72666525019715109346302867952, −2.26165853706545814071229752955, −1.10257519052287726194513676147,
1.10257519052287726194513676147, 2.26165853706545814071229752955, 2.72666525019715109346302867952, 4.31834572542020013709828773845, 4.98408235175799821748161370558, 6.03847146684613022794811719368, 6.31968028107713771021307671812, 7.14175823931883657034929116512, 8.346745577319619020387011706250, 9.076439298074847786705740817908
