| L(s) = 1 | − 7-s + 2·9-s − 4·11-s − 4·23-s − 2·25-s − 4·29-s + 4·37-s − 4·43-s + 49-s + 16·53-s − 2·63-s − 8·67-s + 16·71-s + 4·77-s + 16·79-s − 5·81-s − 8·99-s − 24·107-s − 20·109-s + 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 2/3·9-s − 1.20·11-s − 0.834·23-s − 2/5·25-s − 0.742·29-s + 0.657·37-s − 0.609·43-s + 1/7·49-s + 2.19·53-s − 0.251·63-s − 0.977·67-s + 1.89·71-s + 0.455·77-s + 1.80·79-s − 5/9·81-s − 0.804·99-s − 2.32·107-s − 1.91·109-s + 0.376·113-s − 0.545·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
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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
−8.347573294462718521585261464101, −7.954825299153241619049294570284, −7.71100304878316816193426157991, −7.10806684701676873333766594300, −6.66604112797957105579502813908, −6.18075263653323740699097579635, −5.50672974501345464733363955727, −5.27807460637331501227023990459, −4.60272102909805621149187123785, −3.86124321267387660659658432737, −3.69118644749220381828651256017, −2.61053992400072698195409707231, −2.32663258305922464642586414327, −1.30060073855465746315896674222, 0,
1.30060073855465746315896674222, 2.32663258305922464642586414327, 2.61053992400072698195409707231, 3.69118644749220381828651256017, 3.86124321267387660659658432737, 4.60272102909805621149187123785, 5.27807460637331501227023990459, 5.50672974501345464733363955727, 6.18075263653323740699097579635, 6.66604112797957105579502813908, 7.10806684701676873333766594300, 7.71100304878316816193426157991, 7.954825299153241619049294570284, 8.347573294462718521585261464101
