| L(s) = 1 | − 1.93i·5-s + 3.93i·7-s + 2.03·11-s − 2.46·13-s − 4.76i·17-s + 6.81i·19-s − 7.59·23-s + 1.26·25-s + 2.31i·29-s + 2.87i·31-s + 7.59·35-s − 3.73·37-s − 3.20i·41-s − 3.93i·43-s − 5.56·47-s + ⋯ |
| L(s) = 1 | − 0.863i·5-s + 1.48i·7-s + 0.613·11-s − 0.683·13-s − 1.15i·17-s + 1.56i·19-s − 1.58·23-s + 0.253·25-s + 0.429i·29-s + 0.517i·31-s + 1.28·35-s − 0.613·37-s − 0.500i·41-s − 0.599i·43-s − 0.811·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1324818645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1324818645\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.93iT - 5T^{2} \) |
| 7 | \( 1 - 3.93iT - 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 4.76iT - 17T^{2} \) |
| 19 | \( 1 - 6.81iT - 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 - 2.31iT - 29T^{2} \) |
| 31 | \( 1 - 2.87iT - 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 + 3.20iT - 41T^{2} \) |
| 43 | \( 1 + 3.93iT - 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 5.56T + 59T^{2} \) |
| 61 | \( 1 - 5.73T + 61T^{2} \) |
| 67 | \( 1 - 3.93iT - 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.05iT - 79T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 + 3.62iT - 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−8.566071781981845696292256020368, −8.088617177545803658097135281097, −7.13877529441868573209504785587, −6.30690143218854507869196241485, −5.48872607717653187923670210331, −5.14875133571242962164109900693, −4.19734129953261194955965314164, −3.26542085755795501321347958315, −2.24949131944379505686694395888, −1.50419721924123106474787812325,
0.03414105499779655982440219703, 1.33029211431380271398746108660, 2.44018506500183144216965536872, 3.36194050675563085538246258782, 4.15797797971645075261551575905, 4.63299807758826628464137167368, 5.89510128461360666038709859301, 6.59467465852720617160105618578, 7.06660154881910172592912404716, 7.72401252798482248303514621932
