| L(s) = 1 | + 0.585·5-s − 1.41·7-s − 0.828·11-s − 4.82·13-s + 0.828·17-s − 2.82·19-s + 6.82·23-s − 4.65·25-s + 4.58·29-s + 7.07·31-s − 0.828·35-s − 0.343·37-s − 6.48·41-s + 1.17·43-s − 4.48·47-s − 5·49-s + 10.2·53-s − 0.485·55-s + 9.65·59-s − 11.6·61-s − 2.82·65-s + 5.65·67-s + 8.48·71-s + 11.3·73-s + 1.17·77-s + 14.5·79-s + 3.17·83-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 0.534·7-s − 0.249·11-s − 1.33·13-s + 0.200·17-s − 0.648·19-s + 1.42·23-s − 0.931·25-s + 0.851·29-s + 1.27·31-s − 0.140·35-s − 0.0564·37-s − 1.01·41-s + 0.178·43-s − 0.654·47-s − 0.714·49-s + 1.40·53-s − 0.0654·55-s + 1.25·59-s − 1.49·61-s − 0.350·65-s + 0.691·67-s + 1.00·71-s + 1.32·73-s + 0.133·77-s + 1.64·79-s + 0.348·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.522975890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522975890\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.343T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.273467201549896448495033345867, −7.61143248857725965286825417391, −6.73577780569589316285379961978, −6.33710164187961449297716247683, −5.18893599903686026686783376327, −4.81623986945019954213459661117, −3.69668524116632607588923574462, −2.81186735611702114326456828612, −2.09529422770890324220019873054, −0.66431768037757930688375424781,
0.66431768037757930688375424781, 2.09529422770890324220019873054, 2.81186735611702114326456828612, 3.69668524116632607588923574462, 4.81623986945019954213459661117, 5.18893599903686026686783376327, 6.33710164187961449297716247683, 6.73577780569589316285379961978, 7.61143248857725965286825417391, 8.273467201549896448495033345867
