| L(s) = 1 | − 3·3-s + 5·5-s − 6.80·7-s + 9·9-s − 39.2·11-s − 78.4·13-s − 15·15-s − 95.2·17-s + 133.·19-s + 20.4·21-s + 66.8·23-s + 25·25-s − 27·27-s − 99.6·29-s + 322.·31-s + 117.·33-s − 34.0·35-s − 108.·37-s + 235.·39-s + 278.·41-s − 381.·43-s + 45·45-s + 211.·47-s − 296.·49-s + 285.·51-s + 411.·53-s − 196.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.367·7-s + 0.333·9-s − 1.07·11-s − 1.67·13-s − 0.258·15-s − 1.35·17-s + 1.60·19-s + 0.212·21-s + 0.605·23-s + 0.200·25-s − 0.192·27-s − 0.638·29-s + 1.86·31-s + 0.620·33-s − 0.164·35-s − 0.483·37-s + 0.965·39-s + 1.05·41-s − 1.35·43-s + 0.149·45-s + 0.656·47-s − 0.864·49-s + 0.784·51-s + 1.06·53-s − 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.129702958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129702958\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 6.80T + 343T^{2} \) |
| 11 | \( 1 + 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 322.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 211.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 411.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 447.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 158.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 455.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 630.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 58.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 229.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−9.808267029949172536424856343893, −9.010072674230246781085947443665, −7.75010365016290704387862143411, −7.10401507904464922549496617417, −6.19816434048808493137706004260, −5.14602220073109032400600736801, −4.71407219993688097539027982944, −3.07333217780558377080402980191, −2.18761199407619582808523883114, −0.56335951638783923859999490899,
0.56335951638783923859999490899, 2.18761199407619582808523883114, 3.07333217780558377080402980191, 4.71407219993688097539027982944, 5.14602220073109032400600736801, 6.19816434048808493137706004260, 7.10401507904464922549496617417, 7.75010365016290704387862143411, 9.010072674230246781085947443665, 9.808267029949172536424856343893
