| L(s) = 1 | + 4·2-s + 9·3-s + 26·5-s + 36·6-s − 64·8-s + 104·10-s + 358·11-s − 664·13-s + 234·15-s − 256·16-s + 126·17-s − 2.20e3·19-s + 1.43e3·22-s + 2.14e3·23-s − 576·24-s + 3.12e3·25-s − 2.65e3·26-s − 729·27-s − 7.22e3·29-s + 936·30-s + 5.66e3·31-s + 3.22e3·33-s + 504·34-s + 2.92e3·37-s − 8.80e3·38-s − 5.97e3·39-s − 1.66e3·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.465·5-s + 0.408·6-s − 0.353·8-s + 0.328·10-s + 0.892·11-s − 1.08·13-s + 0.268·15-s − 1/4·16-s + 0.105·17-s − 1.39·19-s + 0.630·22-s + 0.844·23-s − 0.204·24-s + 25-s − 0.770·26-s − 0.192·27-s − 1.59·29-s + 0.189·30-s + 1.05·31-s + 0.515·33-s + 0.0747·34-s + 0.350·37-s − 0.988·38-s − 0.629·39-s − 0.164·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.564917612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.564917612\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 26 T - 2449 T^{2} - 26 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 358 T - 32887 T^{2} - 358 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 332 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 126 T - 1403981 T^{2} - 126 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2200 T + 2363901 T^{2} + 2200 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2142 T - 1848179 T^{2} - 2142 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3610 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5668 T + 3497073 T^{2} - 5668 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2922 T - 60805873 T^{2} - 2922 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2142 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6388 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6520 T - 186834607 T^{2} + 6520 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10702 T - 303662689 T^{2} - 10702 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 42524 T + 1093366277 T^{2} - 42524 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 44840 T + 1166029299 T^{2} + 44840 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1448 T - 1348028403 T^{2} - 1448 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 62 p T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 20500 T - 1652821593 T^{2} - 20500 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 65236 T + 1178679297 T^{2} + 65236 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 102804 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 128006 T + 10801476587 T^{2} + 128006 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 113324 T + p^{5} T^{2} )^{2} \) |
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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
−10.97247486440045530362241191083, −10.95972840037414925701694915414, −10.15188181225150597881321490842, −9.607192464078556703127653208561, −9.364826522892778695543790551364, −8.748365335055392703734936576431, −8.513954658983610688932399918090, −7.73713937743768409212497700691, −7.17846294992893375536625963588, −6.79804518967372025373211861673, −6.05793948831282596266349838456, −5.81949269244628833384782371554, −4.79561554778738951173206647394, −4.74644636969917032823404774823, −3.90387884159142652007025271696, −3.43198829009702849543240627499, −2.54445364563080335554005424394, −2.29781881329715037036176194058, −1.34688145532103827685278266608, −0.49602306900848312024214238818,
0.49602306900848312024214238818, 1.34688145532103827685278266608, 2.29781881329715037036176194058, 2.54445364563080335554005424394, 3.43198829009702849543240627499, 3.90387884159142652007025271696, 4.74644636969917032823404774823, 4.79561554778738951173206647394, 5.81949269244628833384782371554, 6.05793948831282596266349838456, 6.79804518967372025373211861673, 7.17846294992893375536625963588, 7.73713937743768409212497700691, 8.513954658983610688932399918090, 8.748365335055392703734936576431, 9.364826522892778695543790551364, 9.607192464078556703127653208561, 10.15188181225150597881321490842, 10.95972840037414925701694915414, 10.97247486440045530362241191083
