| L(s) = 1 | + 8·7-s − 9-s + 4·17-s − 25-s − 8·31-s − 4·41-s + 16·47-s + 34·49-s − 8·63-s − 20·73-s − 8·79-s + 81-s + 12·89-s − 28·97-s − 8·103-s − 12·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1/3·9-s + 0.970·17-s − 1/5·25-s − 1.43·31-s − 0.624·41-s + 2.33·47-s + 34/7·49-s − 1.00·63-s − 2.34·73-s − 0.900·79-s + 1/9·81-s + 1.27·89-s − 2.84·97-s − 0.788·103-s − 1.12·113-s + 2.93·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.384644811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.384644811\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.456199199943055047372442638182, −8.390808890073914932695668846775, −7.917389547958469343410286500892, −7.64545216559291549965645661336, −7.30330763377889122241515430802, −7.10426317866231359547199814907, −6.47947554495884476747430597346, −5.72295501756643503575230820727, −5.61157920180393600492381051922, −5.43181521849355855509214486042, −4.91378274902303282046666989380, −4.53689416033944674051913400978, −4.09246312450140892214049010299, −3.92604290968544875295448526432, −3.06081657685221378857003155734, −2.77144165431275587296877268361, −1.89402553386274135144197150336, −1.83829720022299553125467197955, −1.30664022616659508182195089341, −0.63368408921395447619768321002,
0.63368408921395447619768321002, 1.30664022616659508182195089341, 1.83829720022299553125467197955, 1.89402553386274135144197150336, 2.77144165431275587296877268361, 3.06081657685221378857003155734, 3.92604290968544875295448526432, 4.09246312450140892214049010299, 4.53689416033944674051913400978, 4.91378274902303282046666989380, 5.43181521849355855509214486042, 5.61157920180393600492381051922, 5.72295501756643503575230820727, 6.47947554495884476747430597346, 7.10426317866231359547199814907, 7.30330763377889122241515430802, 7.64545216559291549965645661336, 7.917389547958469343410286500892, 8.390808890073914932695668846775, 8.456199199943055047372442638182
