L(s) = 1 | − 2·5-s − 2·13-s + 4·17-s + 3·25-s + 12·29-s − 8·31-s − 12·37-s + 4·41-s − 8·43-s − 16·47-s − 6·49-s − 4·53-s − 16·59-s + 12·61-s + 4·65-s − 12·73-s − 16·79-s + 24·83-s − 8·85-s + 20·89-s − 12·97-s − 4·101-s − 8·103-s − 8·107-s + 12·109-s + 4·113-s + 10·121-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.554·13-s + 0.970·17-s + 3/5·25-s + 2.22·29-s − 1.43·31-s − 1.97·37-s + 0.624·41-s − 1.21·43-s − 2.33·47-s − 6/7·49-s − 0.549·53-s − 2.08·59-s + 1.53·61-s + 0.496·65-s − 1.40·73-s − 1.80·79-s + 2.63·83-s − 0.867·85-s + 2.11·89-s − 1.21·97-s − 0.398·101-s − 0.788·103-s − 0.773·107-s + 1.14·109-s + 0.376·113-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.41460912962280748712686776956, −7.35594536902935033431401055348, −6.85880314179626433567342022768, −6.58598012899350398801815035616, −6.22260225012597895047605399714, −5.93531043813178663149838651397, −5.20562818561427680971898818827, −5.13594461422369760381322689657, −4.81824398349804304780095643338, −4.49226605739101475692468494314, −3.98137680303498687551032094866, −3.49539693963193625852063539142, −3.17215271210288275431557873392, −3.16178655158139393482658891511, −2.43111658838752184629167170593, −1.90616292471109375837322076818, −1.43871121741157999331223023008, −1.01736136867745669492980950834, 0, 0,
1.01736136867745669492980950834, 1.43871121741157999331223023008, 1.90616292471109375837322076818, 2.43111658838752184629167170593, 3.16178655158139393482658891511, 3.17215271210288275431557873392, 3.49539693963193625852063539142, 3.98137680303498687551032094866, 4.49226605739101475692468494314, 4.81824398349804304780095643338, 5.13594461422369760381322689657, 5.20562818561427680971898818827, 5.93531043813178663149838651397, 6.22260225012597895047605399714, 6.58598012899350398801815035616, 6.85880314179626433567342022768, 7.35594536902935033431401055348, 7.41460912962280748712686776956