| L(s) = 1 | − 9-s + 4·17-s + 16·23-s + 6·25-s + 16·31-s + 12·41-s − 14·49-s − 16·71-s − 20·73-s − 16·79-s + 81-s + 12·89-s + 4·97-s − 32·103-s + 36·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.970·17-s + 3.33·23-s + 6/5·25-s + 2.87·31-s + 1.87·41-s − 2·49-s − 1.89·71-s − 2.34·73-s − 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.406·97-s − 3.15·103-s + 3.38·113-s + 6/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.325279868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325279868\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 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
−10.53678054538526819429902130252, −10.14143597954171516893615931924, −9.677381846222648336617337407700, −9.221881939788189024996817046142, −8.819304113030716990485867333560, −8.488160237866709217936228452879, −8.009929251330985480190217384238, −7.48771610208422705976173381661, −6.98073821428419389803845495304, −6.79040057983196740369841487635, −5.94958079517156289184029044613, −5.87174871793317313828757853650, −4.95170658338410290830759369963, −4.76249930341247046819410826981, −4.34684880364153666893216696261, −3.29429813718658719386393794195, −2.89013753384545291997686266012, −2.75410683407352004343693392806, −1.34453381360352526103933658464, −0.912968318039718484977089011385,
0.912968318039718484977089011385, 1.34453381360352526103933658464, 2.75410683407352004343693392806, 2.89013753384545291997686266012, 3.29429813718658719386393794195, 4.34684880364153666893216696261, 4.76249930341247046819410826981, 4.95170658338410290830759369963, 5.87174871793317313828757853650, 5.94958079517156289184029044613, 6.79040057983196740369841487635, 6.98073821428419389803845495304, 7.48771610208422705976173381661, 8.009929251330985480190217384238, 8.488160237866709217936228452879, 8.819304113030716990485867333560, 9.221881939788189024996817046142, 9.677381846222648336617337407700, 10.14143597954171516893615931924, 10.53678054538526819429902130252
