L(s) = 1 | + 4·7-s − 4·9-s − 4·11-s − 4·13-s + 2·19-s − 4·23-s − 4·29-s − 8·31-s + 4·37-s − 4·41-s + 12·43-s + 4·47-s + 6·49-s − 4·53-s + 8·59-s − 16·63-s − 8·73-s − 16·77-s + 7·81-s − 20·83-s − 4·89-s − 16·91-s − 12·97-s + 16·99-s − 16·101-s + 24·107-s − 4·109-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 4/3·9-s − 1.20·11-s − 1.10·13-s + 0.458·19-s − 0.834·23-s − 0.742·29-s − 1.43·31-s + 0.657·37-s − 0.624·41-s + 1.82·43-s + 0.583·47-s + 6/7·49-s − 0.549·53-s + 1.04·59-s − 2.01·63-s − 0.936·73-s − 1.82·77-s + 7/9·81-s − 2.19·83-s − 0.423·89-s − 1.67·91-s − 1.21·97-s + 1.60·99-s − 1.59·101-s + 2.32·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 76 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 180 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.76526701851580408560657823519, −7.36512482656878882199443387808, −7.19796501756956204255477343853, −6.86148943881993132966999150927, −6.02897935608791822615126687225, −5.89790640751262423856225591713, −5.43866926626404928728647147993, −5.42806811349004217501668103395, −4.93885171173523565915862087534, −4.63149370091492726338290541486, −4.03957364298085865294994701733, −3.94979543194067834213723320159, −3.13324547354634250800227004362, −2.88370797060470358113630128318, −2.28417353927317556462842344667, −2.28251466495127739814728753537, −1.59640012296732022880350957643, −1.04891637002952638413910268972, 0, 0,
1.04891637002952638413910268972, 1.59640012296732022880350957643, 2.28251466495127739814728753537, 2.28417353927317556462842344667, 2.88370797060470358113630128318, 3.13324547354634250800227004362, 3.94979543194067834213723320159, 4.03957364298085865294994701733, 4.63149370091492726338290541486, 4.93885171173523565915862087534, 5.42806811349004217501668103395, 5.43866926626404928728647147993, 5.89790640751262423856225591713, 6.02897935608791822615126687225, 6.86148943881993132966999150927, 7.19796501756956204255477343853, 7.36512482656878882199443387808, 7.76526701851580408560657823519