| L(s) = 1 | + 6·3-s − 6·5-s + 27·9-s − 2·11-s − 36·15-s − 30·17-s + 12·19-s − 78·23-s − 110·25-s + 108·27-s + 56·29-s + 132·31-s − 12·33-s + 320·37-s − 18·41-s − 376·43-s − 162·45-s − 240·47-s − 180·51-s − 72·53-s + 12·55-s + 72·57-s + 552·59-s + 492·61-s + 540·67-s − 468·69-s − 858·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.536·5-s + 9-s − 0.0548·11-s − 0.619·15-s − 0.428·17-s + 0.144·19-s − 0.707·23-s − 0.879·25-s + 0.769·27-s + 0.358·29-s + 0.764·31-s − 0.0633·33-s + 1.42·37-s − 0.0685·41-s − 1.33·43-s − 0.536·45-s − 0.744·47-s − 0.494·51-s − 0.186·53-s + 0.0294·55-s + 0.167·57-s + 1.21·59-s + 1.03·61-s + 0.984·67-s − 0.816·69-s − 1.43·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.689827898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.689827898\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 1646 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2586 T^{2} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 30 T + 9938 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 12 T + 13302 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 78 T + 16702 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 56 T + 12950 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 132 T + 1698 p T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 320 T + 90294 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 67298 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 188 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 240 T + 193118 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 72 T + 197350 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 552 T + 371222 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 492 T + 492330 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 540 T + 475094 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 858 T + 817486 T^{2} + 858 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 900 T + 943922 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 620 T + 589950 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1224 T + 1402406 T^{2} - 1224 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1422 T + 1792402 T^{2} - 1422 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1116 T + 1756578 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.616264911319696441749326657060, −8.406862629360254394528806746931, −8.103928157901428692543209025358, −7.78755674585841867813738869011, −7.48645561043612994386226060960, −6.92450852338277697168669797679, −6.44131760222373719562418422969, −6.44023679368361240710461843736, −5.55283029209864577739781035708, −5.35595583084138250234693314837, −4.56694956050276462756508106554, −4.43982836179957572352969523131, −3.82108601032252114035091559375, −3.67513050921437170518731192167, −2.82983695103639187657768299317, −2.82921431103529530100965385810, −1.99604107233838231047628937944, −1.75788077709591370422214576444, −0.936793170367792303702327907250, −0.38837998064550755926734499113,
0.38837998064550755926734499113, 0.936793170367792303702327907250, 1.75788077709591370422214576444, 1.99604107233838231047628937944, 2.82921431103529530100965385810, 2.82983695103639187657768299317, 3.67513050921437170518731192167, 3.82108601032252114035091559375, 4.43982836179957572352969523131, 4.56694956050276462756508106554, 5.35595583084138250234693314837, 5.55283029209864577739781035708, 6.44023679368361240710461843736, 6.44131760222373719562418422969, 6.92450852338277697168669797679, 7.48645561043612994386226060960, 7.78755674585841867813738869011, 8.103928157901428692543209025358, 8.406862629360254394528806746931, 8.616264911319696441749326657060
