| L(s) = 1 | − 5-s + 4·7-s + 11-s + 17-s − 6·19-s + 2·23-s + 25-s + 5·31-s − 4·35-s + 2·37-s + 6·41-s + 5·43-s + 4·47-s + 9·49-s + 6·53-s − 55-s + 7·59-s − 8·61-s − 2·67-s + 8·71-s − 7·73-s + 4·77-s − 16·79-s + 9·83-s − 85-s + 9·89-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.242·17-s − 1.37·19-s + 0.417·23-s + 1/5·25-s + 0.898·31-s − 0.676·35-s + 0.328·37-s + 0.937·41-s + 0.762·43-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 0.911·59-s − 1.02·61-s − 0.244·67-s + 0.949·71-s − 0.819·73-s + 0.455·77-s − 1.80·79-s + 0.987·83-s − 0.108·85-s + 0.953·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.513672361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.513672361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.55363008837269, −11.97455671979507, −11.74072230110915, −11.18294363158129, −10.89177845346397, −10.39517797136903, −10.02750815782231, −9.177569594448458, −8.858381110034745, −8.502437268875625, −7.883062336074568, −7.677630748624503, −7.128463235623373, −6.543023042991306, −6.008724639311737, −5.568934630841787, −4.769889244325540, −4.663690213163909, −4.035892935494207, −3.673955305204247, −2.716377279894338, −2.387502031395666, −1.699184601519373, −1.094284826538078, −0.5415359801324787,
0.5415359801324787, 1.094284826538078, 1.699184601519373, 2.387502031395666, 2.716377279894338, 3.673955305204247, 4.035892935494207, 4.663690213163909, 4.769889244325540, 5.568934630841787, 6.008724639311737, 6.543023042991306, 7.128463235623373, 7.677630748624503, 7.883062336074568, 8.502437268875625, 8.858381110034745, 9.177569594448458, 10.02750815782231, 10.39517797136903, 10.89177845346397, 11.18294363158129, 11.74072230110915, 11.97455671979507, 12.55363008837269
