| L(s) = 1 | + 4·2-s + 12·4-s + 7·5-s + 32·8-s + 28·10-s + 25·11-s + 49·13-s + 80·16-s + 98·17-s + 119·19-s + 84·20-s + 100·22-s − 122·23-s − 165·25-s + 196·26-s − 73·29-s − 98·31-s + 192·32-s + 392·34-s + 289·37-s + 476·38-s + 224·40-s + 336·41-s + 307·43-s + 300·44-s − 488·46-s + 672·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.626·5-s + 1.41·8-s + 0.885·10-s + 0.685·11-s + 1.04·13-s + 5/4·16-s + 1.39·17-s + 1.43·19-s + 0.939·20-s + 0.969·22-s − 1.10·23-s − 1.31·25-s + 1.47·26-s − 0.467·29-s − 0.567·31-s + 1.06·32-s + 1.97·34-s + 1.28·37-s + 2.03·38-s + 0.885·40-s + 1.27·41-s + 1.08·43-s + 1.02·44-s − 1.56·46-s + 2.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.46119309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.46119309\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 7 T + 214 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 25 T + 454 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 49 T + 3788 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 98 T + 7402 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 119 T + 16824 T^{2} - 119 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 122 T + 18598 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 73 T + 47746 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 98 T + 24155 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 289 T + 100908 T^{2} - 289 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 672 T + 292750 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 375 T + 141406 T^{2} - 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 763 T + 376762 T^{2} - 763 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 406 T + 439394 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1041 T + 813340 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1652 T + 1360270 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 189 T + 332980 T^{2} - 189 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 524 T + 591329 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 287 T + 781978 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2394 T + 2702050 T^{2} - 2394 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 63 T + 667132 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.03803170006646590271638951988, −9.592210030840439901112085299312, −9.175205876485515002391893933112, −8.857692341334340253098475684309, −7.993192302281939384607014464888, −7.63881009103077876018362556518, −7.46811562653675112526241842432, −6.88576254859943986892018563361, −6.05940811329719390917533505296, −5.98703332250907106742164165093, −5.63256780255610520896533178173, −5.40642832280840336912521422771, −4.38375003381521360112231310915, −4.18644519297324018808516489440, −3.57199122756051470033184270728, −3.34908533914813252848959114073, −2.43574305680165991560157288464, −2.09773004764043409574964812731, −1.20882694562884103306258273392, −0.873397623437876555672511769782,
0.873397623437876555672511769782, 1.20882694562884103306258273392, 2.09773004764043409574964812731, 2.43574305680165991560157288464, 3.34908533914813252848959114073, 3.57199122756051470033184270728, 4.18644519297324018808516489440, 4.38375003381521360112231310915, 5.40642832280840336912521422771, 5.63256780255610520896533178173, 5.98703332250907106742164165093, 6.05940811329719390917533505296, 6.88576254859943986892018563361, 7.46811562653675112526241842432, 7.63881009103077876018362556518, 7.993192302281939384607014464888, 8.857692341334340253098475684309, 9.175205876485515002391893933112, 9.592210030840439901112085299312, 10.03803170006646590271638951988
