L(s) = 1 | + 2·3-s + 3·9-s − 8·11-s + 4·17-s + 8·19-s − 6·25-s + 4·27-s − 16·33-s − 12·41-s − 8·43-s − 14·49-s + 8·51-s + 16·57-s − 8·59-s + 8·67-s + 20·73-s − 12·75-s + 5·81-s + 8·83-s − 12·89-s + 4·97-s − 24·99-s + 24·107-s + 36·113-s + 26·121-s − 24·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 2.41·11-s + 0.970·17-s + 1.83·19-s − 6/5·25-s + 0.769·27-s − 2.78·33-s − 1.87·41-s − 1.21·43-s − 2·49-s + 1.12·51-s + 2.11·57-s − 1.04·59-s + 0.977·67-s + 2.34·73-s − 1.38·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s + 0.406·97-s − 2.41·99-s + 2.32·107-s + 3.38·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.285289264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.285289264\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.58676923620628310825913776398, −11.02059068705223596291987853205, −10.20677110069933956172337352464, −9.785656483951954927896383611288, −9.656797755147918818405236990829, −8.456614023795656741730916500766, −8.173282349834764667604075089133, −7.61947548271568794045481490015, −7.26958585488936859387169650645, −6.16826675668887144123811676692, −5.14444399128618341481533912587, −5.01357758335939619070346679818, −3.43299575130251299450107440988, −3.14826068230388029805668446658, −1.99999348241168201324552330088,
1.99999348241168201324552330088, 3.14826068230388029805668446658, 3.43299575130251299450107440988, 5.01357758335939619070346679818, 5.14444399128618341481533912587, 6.16826675668887144123811676692, 7.26958585488936859387169650645, 7.61947548271568794045481490015, 8.173282349834764667604075089133, 8.456614023795656741730916500766, 9.656797755147918818405236990829, 9.785656483951954927896383611288, 10.20677110069933956172337352464, 11.02059068705223596291987853205, 11.58676923620628310825913776398