L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 3·9-s + 2·11-s − 2·13-s − 4·15-s + 6·17-s − 4·21-s + 3·25-s − 4·27-s + 16·29-s + 4·31-s − 4·33-s + 4·35-s + 8·37-s + 4·39-s + 16·41-s − 14·43-s + 6·45-s + 4·47-s + 2·49-s − 12·51-s + 4·53-s + 4·55-s + 16·59-s + 6·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s + 1.45·17-s − 0.872·21-s + 3/5·25-s − 0.769·27-s + 2.97·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s + 1.31·37-s + 0.640·39-s + 2.49·41-s − 2.13·43-s + 0.894·45-s + 0.583·47-s + 2/7·49-s − 1.68·51-s + 0.549·53-s + 0.539·55-s + 2.08·59-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071101820\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071101820\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 202 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.47759555204823211682624603074, −10.37631403991110681227434958537, −10.03021810875603782722581493037, −9.672553047505704057353303332034, −9.077332575650518825471885753296, −8.631126164531151275795714780150, −7.968445159272757478837738916360, −7.80785539785542454345538288205, −6.98926022453990045871448951669, −6.75249812853776530673041676529, −6.10370565275214544777301635978, −5.91546859152013761914224244641, −5.23006816340440524352163590418, −4.97062700541005579419933010852, −4.38058532201063213363452658859, −3.94652552188525381019698908946, −2.82041330729035525767727240259, −2.50369511805363283295151085528, −1.26358888529032209437631311984, −1.05741522491710042055576261306,
1.05741522491710042055576261306, 1.26358888529032209437631311984, 2.50369511805363283295151085528, 2.82041330729035525767727240259, 3.94652552188525381019698908946, 4.38058532201063213363452658859, 4.97062700541005579419933010852, 5.23006816340440524352163590418, 5.91546859152013761914224244641, 6.10370565275214544777301635978, 6.75249812853776530673041676529, 6.98926022453990045871448951669, 7.80785539785542454345538288205, 7.968445159272757478837738916360, 8.631126164531151275795714780150, 9.077332575650518825471885753296, 9.672553047505704057353303332034, 10.03021810875603782722581493037, 10.37631403991110681227434958537, 10.47759555204823211682624603074