| L(s) = 1 | + 42·3-s − 4·7-s + 1.03e3·9-s + 5.90e3·13-s + 1.05e4·19-s − 168·21-s + 2.45e3·25-s + 1.28e4·27-s − 4.57e4·31-s − 6.81e4·37-s + 2.47e5·39-s − 1.28e4·43-s − 2.35e5·49-s + 4.41e5·57-s + 1.25e5·61-s − 4.14e3·63-s + 8.77e5·67-s − 1.46e6·73-s + 1.02e5·75-s − 6.81e5·79-s − 2.14e5·81-s − 2.36e4·91-s − 1.92e6·93-s − 5.62e5·97-s + 1.73e6·103-s − 1.30e6·109-s − 2.86e6·111-s + ⋯ |
| L(s) = 1 | + 14/9·3-s − 0.0116·7-s + 1.41·9-s + 2.68·13-s + 1.53·19-s − 0.0181·21-s + 0.156·25-s + 0.652·27-s − 1.53·31-s − 1.34·37-s + 4.17·39-s − 0.161·43-s − 1.99·49-s + 2.38·57-s + 0.551·61-s − 0.0165·63-s + 2.91·67-s − 3.75·73-s + 0.243·75-s − 1.38·79-s − 0.404·81-s − 0.0313·91-s − 2.39·93-s − 0.615·97-s + 1.58·103-s − 1.00·109-s − 2.09·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(7.026186105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.026186105\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 - 14 p T + p^{6} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 98 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3541970 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2950 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 28202690 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5258 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 191004770 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1184779442 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 22898 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34058 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9219079010 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6406 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10801249342 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7253988050 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22449655150 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 62566 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 438698 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 251546372642 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 730510 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 340562 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 407613512306 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 844406214050 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 281086 T + p^{6} T^{2} )^{2} \) |
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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
−11.45697497139952927510399688487, −11.27931226073062305070481924154, −10.74368596418649144942141570599, −9.978846192410678693398586551817, −9.701778127534721578996932972394, −8.855678535000813679410146879480, −8.853973201526653298026096295689, −8.249987926769466297151979920857, −7.84691614615780861727842769485, −7.08467390358927661683014423737, −6.77972713976186605829136548090, −5.78661164662622800158483702564, −5.54211654068145731163654578259, −4.48888631046966488275373295392, −3.81389705841768843819329426898, −3.20465035657406611273763307849, −3.18237039648430064321025858981, −1.78630918300360921186585079824, −1.58599669465352602166636390188, −0.67386903621030821180945764961,
0.67386903621030821180945764961, 1.58599669465352602166636390188, 1.78630918300360921186585079824, 3.18237039648430064321025858981, 3.20465035657406611273763307849, 3.81389705841768843819329426898, 4.48888631046966488275373295392, 5.54211654068145731163654578259, 5.78661164662622800158483702564, 6.77972713976186605829136548090, 7.08467390358927661683014423737, 7.84691614615780861727842769485, 8.249987926769466297151979920857, 8.853973201526653298026096295689, 8.855678535000813679410146879480, 9.701778127534721578996932972394, 9.978846192410678693398586551817, 10.74368596418649144942141570599, 11.27931226073062305070481924154, 11.45697497139952927510399688487
