L(s) = 1 | + 2·2-s + 4-s − 2·8-s + 6·9-s − 2·11-s − 4·16-s + 12·18-s − 4·22-s + 16·23-s + 2·25-s − 24·29-s − 2·32-s + 6·36-s + 12·37-s − 16·43-s − 2·44-s + 32·46-s + 4·50-s − 12·53-s − 48·58-s + 3·64-s + 8·67-s − 12·72-s + 24·74-s + 9·81-s − 32·86-s + 4·88-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 2·9-s − 0.603·11-s − 16-s + 2.82·18-s − 0.852·22-s + 3.33·23-s + 2/5·25-s − 4.45·29-s − 0.353·32-s + 36-s + 1.97·37-s − 2.43·43-s − 0.301·44-s + 4.71·46-s + 0.565·50-s − 1.64·53-s − 6.30·58-s + 3/8·64-s + 0.977·67-s − 1.41·72-s + 2.78·74-s + 81-s − 3.45·86-s + 0.426·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.280791202\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.280791202\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 10 T^{2} - 861 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 86 T^{2} + 3915 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 90 T^{2} + 4379 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 74 T^{2} + 147 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 50 T^{2} - 5421 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.16120151184465327456370574991, −6.76938898477563263820843820119, −6.68324304966303591935851925566, −6.25848922324922639371056781300, −6.14067038847960099175490482795, −6.05877984222746280942824781434, −5.51635620883250021461696594642, −5.19676069519155742663650157149, −5.11379231616085763030492813835, −5.10612460542246050793630312123, −5.06729461168235260334755207169, −4.32397177036322888782790060971, −4.21049040414173872777529111635, −4.12232062349114558505441557990, −4.10347088674538675424327364193, −3.40538605827321420226286436414, −3.17528496827310334658313984916, −3.17192163934378238436428927005, −2.99263862540004565990137327816, −2.41418343712792240296525227159, −1.86475686117657872900134354658, −1.85296439019128678380914294731, −1.52711398307412433780369009222, −0.914690259957922467532406923471, −0.47091112443030590078212435359,
0.47091112443030590078212435359, 0.914690259957922467532406923471, 1.52711398307412433780369009222, 1.85296439019128678380914294731, 1.86475686117657872900134354658, 2.41418343712792240296525227159, 2.99263862540004565990137327816, 3.17192163934378238436428927005, 3.17528496827310334658313984916, 3.40538605827321420226286436414, 4.10347088674538675424327364193, 4.12232062349114558505441557990, 4.21049040414173872777529111635, 4.32397177036322888782790060971, 5.06729461168235260334755207169, 5.10612460542246050793630312123, 5.11379231616085763030492813835, 5.19676069519155742663650157149, 5.51635620883250021461696594642, 6.05877984222746280942824781434, 6.14067038847960099175490482795, 6.25848922324922639371056781300, 6.68324304966303591935851925566, 6.76938898477563263820843820119, 7.16120151184465327456370574991