| L(s) = 1 | − 4·4-s + 12·9-s + 12·16-s + 20·25-s − 48·36-s + 24·49-s − 8·53-s − 32·64-s + 90·81-s − 80·100-s − 80·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 144·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4·9-s + 3·16-s + 4·25-s − 8·36-s + 24/7·49-s − 1.09·53-s − 4·64-s + 10·81-s − 8·100-s − 7.52·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 12·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.073005031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.073005031\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.905086591732308789496925160083, −7.48853028323241078581961901640, −7.20631296219976137619443419523, −6.94116333386217245079951902371, −6.93197087331034927211459175589, −6.62138028648184619511750863691, −6.58815029907061836577563607635, −5.97467967240815976958241322479, −5.70650390375927511866652443442, −5.28683802006174434235050073576, −5.27826290059590094971809743771, −4.88544629810239179545785147350, −4.58863696712385945258153389738, −4.47566090405179387501475894352, −4.40700961180618654895103713724, −3.93269919705747054175798928512, −3.89485313292287050871122019952, −3.39559670026226394060869617387, −3.29402906150491767102777512644, −2.52981946416455970061037688243, −2.51276715607953967433012935953, −1.64353582131577118076378563720, −1.27546066207092635188120514625, −1.12871449318928671388025298562, −0.73676909444637053229286842711,
0.73676909444637053229286842711, 1.12871449318928671388025298562, 1.27546066207092635188120514625, 1.64353582131577118076378563720, 2.51276715607953967433012935953, 2.52981946416455970061037688243, 3.29402906150491767102777512644, 3.39559670026226394060869617387, 3.89485313292287050871122019952, 3.93269919705747054175798928512, 4.40700961180618654895103713724, 4.47566090405179387501475894352, 4.58863696712385945258153389738, 4.88544629810239179545785147350, 5.27826290059590094971809743771, 5.28683802006174434235050073576, 5.70650390375927511866652443442, 5.97467967240815976958241322479, 6.58815029907061836577563607635, 6.62138028648184619511750863691, 6.93197087331034927211459175589, 6.94116333386217245079951902371, 7.20631296219976137619443419523, 7.48853028323241078581961901640, 7.905086591732308789496925160083
