| L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s + 16-s + 6·17-s − 2·18-s − 2·25-s − 32-s − 6·34-s + 2·36-s + 14·49-s + 2·50-s + 64-s + 6·68-s − 2·72-s − 5·81-s + 12·89-s − 14·98-s − 2·100-s − 16·103-s − 14·121-s + 127-s − 128-s + 131-s − 6·136-s + 137-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1/4·16-s + 1.45·17-s − 0.471·18-s − 2/5·25-s − 0.176·32-s − 1.02·34-s + 1/3·36-s + 2·49-s + 0.282·50-s + 1/8·64-s + 0.727·68-s − 0.235·72-s − 5/9·81-s + 1.27·89-s − 1.41·98-s − 1/5·100-s − 1.57·103-s − 1.27·121-s + 0.0887·127-s − 0.0883·128-s + 0.0873·131-s − 0.514·136-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014991940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014991940\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.20577814736085356952207776211, −9.884387289899397217041620201286, −9.364746929225894270075652617230, −8.835863216401317832866339248203, −8.190590441387193158826838716515, −7.75591260690702797156496136088, −7.21242822907521813396765907095, −6.79518706283797008668965856547, −5.89942135419593380940820484985, −5.57701141450097199376705821999, −4.68351373799807119232126686211, −3.90867162313150085392849000020, −3.19825512904115960807108105793, −2.20924692317133984874547287656, −1.12942124871842761561442350612,
1.12942124871842761561442350612, 2.20924692317133984874547287656, 3.19825512904115960807108105793, 3.90867162313150085392849000020, 4.68351373799807119232126686211, 5.57701141450097199376705821999, 5.89942135419593380940820484985, 6.79518706283797008668965856547, 7.21242822907521813396765907095, 7.75591260690702797156496136088, 8.190590441387193158826838716515, 8.835863216401317832866339248203, 9.364746929225894270075652617230, 9.884387289899397217041620201286, 10.20577814736085356952207776211
