| L(s) = 1 | + 1.98·2-s − 3-s + 1.93·4-s + 5-s − 1.98·6-s + 1.53·7-s − 0.121·8-s + 9-s + 1.98·10-s + 3.86·11-s − 1.93·12-s − 3.56·13-s + 3.04·14-s − 15-s − 4.11·16-s − 6.61·17-s + 1.98·18-s + 5.57·19-s + 1.93·20-s − 1.53·21-s + 7.66·22-s + 0.121·24-s + 25-s − 7.08·26-s − 27-s + 2.97·28-s + 7.13·29-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.969·4-s + 0.447·5-s − 0.810·6-s + 0.580·7-s − 0.0429·8-s + 0.333·9-s + 0.627·10-s + 1.16·11-s − 0.559·12-s − 0.989·13-s + 0.814·14-s − 0.258·15-s − 1.02·16-s − 1.60·17-s + 0.467·18-s + 1.27·19-s + 0.433·20-s − 0.335·21-s + 1.63·22-s + 0.0247·24-s + 0.200·25-s − 1.38·26-s − 0.192·27-s + 0.562·28-s + 1.32·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.203697244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.203697244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 + 4.05T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 6.28T + 97T^{2} \) |
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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
−7.41949906755661146720925823390, −6.87689934327789408766843009128, −6.32316248195280860783322627939, −5.53129043144363734955725677760, −5.07459740281513028139774097735, −4.34686241031703831479187931991, −3.89596643432538651113332246060, −2.71610506917227497440462812775, −2.07754564548715952049926113947, −0.853729479839237237251584357260,
0.853729479839237237251584357260, 2.07754564548715952049926113947, 2.71610506917227497440462812775, 3.89596643432538651113332246060, 4.34686241031703831479187931991, 5.07459740281513028139774097735, 5.53129043144363734955725677760, 6.32316248195280860783322627939, 6.87689934327789408766843009128, 7.41949906755661146720925823390
