| L(s) = 1 | + 2·5-s + 2·7-s − 2·11-s + 13-s − 12·17-s − 12·19-s + 8·23-s + 5·25-s + 2·29-s − 10·31-s + 4·35-s − 12·37-s − 6·41-s − 4·43-s − 2·47-s + 7·49-s − 12·53-s − 4·55-s − 10·59-s + 2·61-s + 2·65-s − 10·67-s − 20·71-s + 4·73-s − 4·77-s + 4·79-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.603·11-s + 0.277·13-s − 2.91·17-s − 2.75·19-s + 1.66·23-s + 25-s + 0.371·29-s − 1.79·31-s + 0.676·35-s − 1.97·37-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 49-s − 1.64·53-s − 0.539·55-s − 1.30·59-s + 0.256·61-s + 0.248·65-s − 1.22·67-s − 2.37·71-s + 0.468·73-s − 0.455·77-s + 0.450·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.481251033826456252685701795446, −7.935402086638341424766671292984, −7.29435072098495924530384693622, −7.13960985732195713161698660566, −6.63221975556018327223869234947, −6.51295219091066507792724493691, −6.07062420617015257892271551804, −5.70292559592217599261584780939, −4.96715169147961621429874312548, −4.94846342074959796905846944028, −4.48793479799210210532315241885, −4.32434264461534426660229144821, −3.46544225614783731464777974699, −3.22416581993473666532195864540, −2.33658408283403394655731361427, −2.30304545456749017067089545834, −1.73239042129283392178990796261, −1.44848883123461178262717839400, 0, 0,
1.44848883123461178262717839400, 1.73239042129283392178990796261, 2.30304545456749017067089545834, 2.33658408283403394655731361427, 3.22416581993473666532195864540, 3.46544225614783731464777974699, 4.32434264461534426660229144821, 4.48793479799210210532315241885, 4.94846342074959796905846944028, 4.96715169147961621429874312548, 5.70292559592217599261584780939, 6.07062420617015257892271551804, 6.51295219091066507792724493691, 6.63221975556018327223869234947, 7.13960985732195713161698660566, 7.29435072098495924530384693622, 7.935402086638341424766671292984, 8.481251033826456252685701795446
