| L(s) = 1 | + 12·5-s − 32·7-s + 8·11-s + 20·13-s + 98·17-s − 88·19-s − 32·23-s + 19·25-s + 172·29-s + 256·31-s − 384·35-s − 92·37-s − 102·41-s − 296·43-s − 320·47-s + 681·49-s + 76·53-s + 96·55-s − 408·59-s − 636·61-s + 240·65-s + 552·67-s + 416·71-s + 138·73-s − 256·77-s + 64·79-s − 392·83-s + ⋯ |
| L(s) = 1 | + 1.07·5-s − 1.72·7-s + 0.219·11-s + 0.426·13-s + 1.39·17-s − 1.06·19-s − 0.290·23-s + 0.151·25-s + 1.10·29-s + 1.48·31-s − 1.85·35-s − 0.408·37-s − 0.388·41-s − 1.04·43-s − 0.993·47-s + 1.98·49-s + 0.196·53-s + 0.235·55-s − 0.900·59-s − 1.33·61-s + 0.457·65-s + 1.00·67-s + 0.695·71-s + 0.221·73-s − 0.378·77-s + 0.0911·79-s − 0.518·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.122059318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122059318\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 98 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 32 T + p^{3} T^{2} \) |
| 29 | \( 1 - 172 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 92 T + p^{3} T^{2} \) |
| 41 | \( 1 + 102 T + p^{3} T^{2} \) |
| 43 | \( 1 + 296 T + p^{3} T^{2} \) |
| 47 | \( 1 + 320 T + p^{3} T^{2} \) |
| 53 | \( 1 - 76 T + p^{3} T^{2} \) |
| 59 | \( 1 + 408 T + p^{3} T^{2} \) |
| 61 | \( 1 + 636 T + p^{3} T^{2} \) |
| 67 | \( 1 - 552 T + p^{3} T^{2} \) |
| 71 | \( 1 - 416 T + p^{3} T^{2} \) |
| 73 | \( 1 - 138 T + p^{3} T^{2} \) |
| 79 | \( 1 - 64 T + p^{3} T^{2} \) |
| 83 | \( 1 + 392 T + p^{3} T^{2} \) |
| 89 | \( 1 - 582 T + p^{3} T^{2} \) |
| 97 | \( 1 - 238 T + p^{3} T^{2} \) |
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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
−8.772412615026679372098213662253, −7.981824826142754786051355403594, −6.74384096701357001240943659123, −6.32853266863734027870245179640, −5.80120999958457815903300391722, −4.71497115981401218231263869910, −3.53226189971932062698429342745, −2.93168412979208911012812133740, −1.82227113330405249333578235128, −0.64553909414137209893317528988,
0.64553909414137209893317528988, 1.82227113330405249333578235128, 2.93168412979208911012812133740, 3.53226189971932062698429342745, 4.71497115981401218231263869910, 5.80120999958457815903300391722, 6.32853266863734027870245179640, 6.74384096701357001240943659123, 7.981824826142754786051355403594, 8.772412615026679372098213662253
