| L(s) = 1 | − 8·3-s + 24·9-s − 24·11-s + 32·17-s − 56·19-s + 60·25-s − 40·27-s + 192·33-s − 56·41-s − 200·43-s + 20·49-s − 256·51-s + 448·57-s − 40·59-s − 88·67-s − 112·73-s − 480·75-s + 110·81-s + 152·83-s + 144·89-s − 576·99-s + 232·107-s − 168·113-s − 104·121-s + 448·123-s + 127-s + 1.60e3·129-s + ⋯ |
| L(s) = 1 | − 8/3·3-s + 8/3·9-s − 2.18·11-s + 1.88·17-s − 2.94·19-s + 12/5·25-s − 1.48·27-s + 5.81·33-s − 1.36·41-s − 4.65·43-s + 0.408·49-s − 5.01·51-s + 7.85·57-s − 0.677·59-s − 1.31·67-s − 1.53·73-s − 6.39·75-s + 1.35·81-s + 1.83·83-s + 1.61·89-s − 5.81·99-s + 2.16·107-s − 1.48·113-s − 0.859·121-s + 3.64·123-s + 0.00787·127-s + 12.4·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1255379528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1255379528\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T + 4 p T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 6 p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2342 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 268 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 380 T^{2} + 82982 T^{4} - 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 16 T + 482 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 28 T + 668 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1556 T^{2} + 1100966 T^{4} - 1556 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1340 T^{2} + 839462 T^{4} - 1340 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1604 T^{2} + 1466246 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 380 T^{2} + 2545382 T^{4} - 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 998 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 100 T + 6188 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2180 T^{2} + 461702 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4796 T^{2} + 7574 p^{2} T^{4} - 4796 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 3452 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 9340 T^{2} + 42035622 T^{4} - 9340 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 44 T + 3212 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9620 T^{2} + 52760102 T^{4} - 9620 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 56 T + 11282 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3900 T^{2} + 63639302 T^{4} + 3900 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 76 T + 14012 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 72 T + 15698 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18658 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.63740494754141074450548627058, −7.38544734818080289238573649123, −7.09182865507764577535409243409, −6.64716328248311086935732348851, −6.45835720807561059855182165646, −6.42789022663579642612739856407, −6.35585199884947988723512126322, −5.79348519744697355605029860128, −5.56864559806171853931003109847, −5.36789051931033934616767596854, −5.35491460583410003922732510976, −4.94859665854129489531391537718, −4.81351296860041616691668909437, −4.50379987637622544147974363119, −4.40817431820289703457805086728, −3.68282280926532228832328193913, −3.32828432706658204604200077586, −3.26865744297654491128302212159, −2.85214167536015755698933423218, −2.44655532140887483053882497891, −1.87802590936530840419979385312, −1.73509008651600850963567154213, −1.08901343037642558816140454220, −0.42950645054083899442897176541, −0.17572894634675579883839228214,
0.17572894634675579883839228214, 0.42950645054083899442897176541, 1.08901343037642558816140454220, 1.73509008651600850963567154213, 1.87802590936530840419979385312, 2.44655532140887483053882497891, 2.85214167536015755698933423218, 3.26865744297654491128302212159, 3.32828432706658204604200077586, 3.68282280926532228832328193913, 4.40817431820289703457805086728, 4.50379987637622544147974363119, 4.81351296860041616691668909437, 4.94859665854129489531391537718, 5.35491460583410003922732510976, 5.36789051931033934616767596854, 5.56864559806171853931003109847, 5.79348519744697355605029860128, 6.35585199884947988723512126322, 6.42789022663579642612739856407, 6.45835720807561059855182165646, 6.64716328248311086935732348851, 7.09182865507764577535409243409, 7.38544734818080289238573649123, 7.63740494754141074450548627058
