| L(s) = 1 | − 27·3-s + 100·5-s − 2.77e3·11-s + 6.58e3·13-s − 2.70e3·15-s + 5.90e3·17-s + 6.64e3·19-s − 1.98e3·23-s + 7.81e4·25-s + 1.96e4·27-s − 4.16e5·29-s − 1.17e5·31-s + 7.48e4·33-s + 3.35e5·37-s − 1.77e5·39-s + 5.30e5·41-s − 1.86e5·43-s − 6.57e5·47-s − 1.59e5·51-s + 6.08e5·53-s − 2.77e5·55-s − 1.79e5·57-s − 5.36e5·59-s − 1.79e6·61-s + 6.58e5·65-s − 2.12e6·67-s + 5.35e4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.357·5-s − 0.628·11-s + 0.831·13-s − 0.206·15-s + 0.291·17-s + 0.222·19-s − 0.0339·23-s + 25-s + 0.192·27-s − 3.16·29-s − 0.710·31-s + 0.362·33-s + 1.08·37-s − 0.480·39-s + 1.20·41-s − 0.357·43-s − 0.923·47-s − 0.168·51-s + 0.561·53-s − 0.224·55-s − 0.128·57-s − 0.339·59-s − 1.01·61-s + 0.297·65-s − 0.862·67-s + 0.0196·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7057828662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7057828662\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 p^{2} T - 109 p^{4} T^{2} - 4 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2774 T - 11792095 T^{2} + 2774 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3294 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5900 T - 375528673 T^{2} - 5900 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6644 T - 849729003 T^{2} - 6644 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1982 T - 3400897123 T^{2} + 1982 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 208106 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 117792 T - 13637658847 T^{2} + 117792 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 335686 T + 17753213463 T^{2} - 335686 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 265488 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 93292 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 657516 T - 74295830207 T^{2} + 657516 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 608718 T - 804173536313 T^{2} - 608718 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 536120 T - 2201226830419 T^{2} + 536120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1797090 T + 86789632079 T^{2} + 1797090 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2123176 T - 1552835278347 T^{2} + 2123176 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1191214 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1056430 T - 9931354174197 T^{2} - 1056430 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 998484 T - 18206938687903 T^{2} + 998484 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3898004 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4622352 T - 22865196883625 T^{2} + 4622352 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15287710 T + p^{7} T^{2} )^{2} \) |
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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
−9.625381915387839157045186062781, −9.551411826891306780185316463456, −8.885573298601572193733504077793, −8.604843249529118152959549252229, −7.900890635106704291208135095297, −7.57297566452261101417589658592, −7.16429055439546099825299757029, −6.64045127505009936824922357413, −5.96875205420502868933299406361, −5.70192651378745701886026809143, −5.47837886231782694960400297779, −4.81690428332309209732710504716, −4.21334224100799842892188565284, −3.79896731274207325923940704881, −3.06666647050749870565052171681, −2.75953644870299070314037985590, −1.85454728327641475132521290061, −1.55378111029448780272211417178, −0.891683237118341245798690430846, −0.18476175252631312872971162899,
0.18476175252631312872971162899, 0.891683237118341245798690430846, 1.55378111029448780272211417178, 1.85454728327641475132521290061, 2.75953644870299070314037985590, 3.06666647050749870565052171681, 3.79896731274207325923940704881, 4.21334224100799842892188565284, 4.81690428332309209732710504716, 5.47837886231782694960400297779, 5.70192651378745701886026809143, 5.96875205420502868933299406361, 6.64045127505009936824922357413, 7.16429055439546099825299757029, 7.57297566452261101417589658592, 7.900890635106704291208135095297, 8.604843249529118152959549252229, 8.885573298601572193733504077793, 9.551411826891306780185316463456, 9.625381915387839157045186062781
