| L(s) = 1 | − 3-s + 6·5-s + 6·11-s − 6·15-s + 12·17-s − 7·19-s + 19·25-s + 27-s − 5·31-s − 6·33-s − 37-s + 6·47-s − 12·51-s + 36·55-s + 7·57-s − 3·67-s − 15·73-s − 19·75-s + 27·79-s − 81-s − 12·83-s + 72·85-s − 12·89-s + 5·93-s − 42·95-s + 6·101-s − 5·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s + 1.80·11-s − 1.54·15-s + 2.91·17-s − 1.60·19-s + 19/5·25-s + 0.192·27-s − 0.898·31-s − 1.04·33-s − 0.164·37-s + 0.875·47-s − 1.68·51-s + 4.85·55-s + 0.927·57-s − 0.366·67-s − 1.75·73-s − 2.19·75-s + 3.03·79-s − 1/9·81-s − 1.31·83-s + 7.80·85-s − 1.27·89-s + 0.518·93-s − 4.30·95-s + 0.597·101-s − 0.492·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.972769216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.972769216\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.252450178627002607378927750841, −9.063612335781129426339583293787, −8.402476117195128610810920786112, −8.264499095423831681303709712487, −7.31937147319486554677172043508, −7.31702374789954103649794288886, −6.55240958564517376145313109432, −6.34298678050721588464292886384, −6.09697011478991417040065035668, −5.62479475580597727560957611375, −5.44102065952285412942400214704, −5.15536397300438695115661271921, −4.19753776495038422714554256498, −4.18669324297706542238677364867, −3.15220864268317864617552025386, −3.13795774454789076896795800691, −2.06251900713278261862893669241, −1.91986960575059994913610143153, −1.32906963022462873338354535929, −0.863161756554439358181183586219,
0.863161756554439358181183586219, 1.32906963022462873338354535929, 1.91986960575059994913610143153, 2.06251900713278261862893669241, 3.13795774454789076896795800691, 3.15220864268317864617552025386, 4.18669324297706542238677364867, 4.19753776495038422714554256498, 5.15536397300438695115661271921, 5.44102065952285412942400214704, 5.62479475580597727560957611375, 6.09697011478991417040065035668, 6.34298678050721588464292886384, 6.55240958564517376145313109432, 7.31702374789954103649794288886, 7.31937147319486554677172043508, 8.264499095423831681303709712487, 8.402476117195128610810920786112, 9.063612335781129426339583293787, 9.252450178627002607378927750841
