| 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
| \(L(2)\) |
\(\approx\) |
\(4.304644502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.304644502\) |
| \(L(\frac{5}{2})\) |
|
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
−8.717379175020210442957901903024, −7.76888128119596904717117358372, −7.37526643763775611598722775316, −6.08024105872374638682638685584, −5.41695663587065705126482843747, −4.96063462717509695870465451092, −3.79356918589655850558880033858, −2.65638059867526399040251049309, −1.63874273791793749346484715843, −1.05450081924230064004478520222,
1.05450081924230064004478520222, 1.63874273791793749346484715843, 2.65638059867526399040251049309, 3.79356918589655850558880033858, 4.96063462717509695870465451092, 5.41695663587065705126482843747, 6.08024105872374638682638685584, 7.37526643763775611598722775316, 7.76888128119596904717117358372, 8.717379175020210442957901903024
