| L(s) = 1 | + 3·7-s − 7·13-s − 5·25-s + 18·31-s − 2·37-s + 18·43-s − 49-s + 13·61-s + 21·67-s − 34·73-s − 21·79-s − 21·91-s + 5·97-s + 33·103-s − 4·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.94·13-s − 25-s + 3.23·31-s − 0.328·37-s + 2.74·43-s − 1/7·49-s + 1.66·61-s + 2.56·67-s − 3.97·73-s − 2.36·79-s − 2.20·91-s + 0.507·97-s + 3.25·103-s − 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.145274172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145274172\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.987931261664538455981097207418, −9.557008989735424653239844180729, −8.961323336091691277515780311441, −8.637402297087021235465809745787, −8.175405868009276964261154139925, −7.84032270905935207785416255930, −7.39027513940284483757819696950, −7.21945113043811537508773283024, −6.58092812848564055919857418870, −6.08088428300557787770974127448, −5.64841656381367858398908003175, −5.18188997729255598737337648950, −4.57676811708720689916874418388, −4.52279972003395149616113414066, −3.99183490254623687266768140380, −3.11839595404436634647512563178, −2.48779330918976531059836383661, −2.33102846419079913114249667782, −1.44303450871384014936075293059, −0.63325911788407493998414587520,
0.63325911788407493998414587520, 1.44303450871384014936075293059, 2.33102846419079913114249667782, 2.48779330918976531059836383661, 3.11839595404436634647512563178, 3.99183490254623687266768140380, 4.52279972003395149616113414066, 4.57676811708720689916874418388, 5.18188997729255598737337648950, 5.64841656381367858398908003175, 6.08088428300557787770974127448, 6.58092812848564055919857418870, 7.21945113043811537508773283024, 7.39027513940284483757819696950, 7.84032270905935207785416255930, 8.175405868009276964261154139925, 8.637402297087021235465809745787, 8.961323336091691277515780311441, 9.557008989735424653239844180729, 9.987931261664538455981097207418
