| L(s) = 1 | − 3-s + 2.70·5-s + 7-s + 9-s − 0.701·13-s − 2.70·15-s − 7.40·19-s − 21-s + 23-s + 2.29·25-s − 27-s + 6.70·29-s − 6·31-s + 2.70·35-s − 2.70·37-s + 0.701·39-s + 1.29·41-s + 0.701·43-s + 2.70·45-s + 2.70·47-s + 49-s − 3.40·53-s + 7.40·57-s + 13.4·59-s + 10·61-s + 63-s − 1.89·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.20·5-s + 0.377·7-s + 0.333·9-s − 0.194·13-s − 0.697·15-s − 1.69·19-s − 0.218·21-s + 0.208·23-s + 0.459·25-s − 0.192·27-s + 1.24·29-s − 1.07·31-s + 0.456·35-s − 0.444·37-s + 0.112·39-s + 0.202·41-s + 0.106·43-s + 0.402·45-s + 0.394·47-s + 0.142·49-s − 0.467·53-s + 0.980·57-s + 1.74·59-s + 1.28·61-s + 0.125·63-s − 0.235·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068366090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068366090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.40T + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 - 0.701T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.86276839580977946111347216892, −6.92304495735083127179501299359, −6.44155989087109137977498611470, −5.78604918361151373628413538569, −5.14747000277713244781728414417, −4.50773949630230977089631940320, −3.60510800204143525155344879654, −2.34019664005789748968242166072, −1.91575197038131678670837107746, −0.73625241966691645276230774570,
0.73625241966691645276230774570, 1.91575197038131678670837107746, 2.34019664005789748968242166072, 3.60510800204143525155344879654, 4.50773949630230977089631940320, 5.14747000277713244781728414417, 5.78604918361151373628413538569, 6.44155989087109137977498611470, 6.92304495735083127179501299359, 7.86276839580977946111347216892
