| L(s) = 1 | − 2-s + 5-s + 2·9-s − 10-s − 11-s + 13-s − 2·18-s + 19-s + 22-s − 23-s − 26-s + 31-s + 32-s − 38-s + 2·45-s + 46-s − 55-s − 62-s − 64-s + 65-s − 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s + 89-s − 2·90-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 2·9-s − 10-s − 11-s + 13-s − 2·18-s + 19-s + 22-s − 23-s − 26-s + 31-s + 32-s − 38-s + 2·45-s + 46-s − 55-s − 62-s − 64-s + 65-s − 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s + 89-s − 2·90-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14984641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14984641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.093707504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093707504\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 79 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
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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
−8.934058520122654096880560447025, −8.483752216984281511831028398115, −8.087689005730467510298560007433, −7.86296963177424278798038184268, −7.52444368932463856971907750757, −7.02303209470808462205207979005, −6.76349931858681190969506761185, −6.20101623033584227621280417147, −6.02293828762766177802572178255, −5.61619938303031494470719440714, −4.97724897268879851068015874740, −4.84582754058516930157914867655, −4.18193711501116236739734378419, −4.00197664824122313983798470526, −3.31034910086610453511947107938, −2.90615668789877121469611312661, −2.21379826729241298497499007127, −1.86071031013832090378576423191, −1.29119035466535395828572803795, −0.795087921720937410396009565130,
0.795087921720937410396009565130, 1.29119035466535395828572803795, 1.86071031013832090378576423191, 2.21379826729241298497499007127, 2.90615668789877121469611312661, 3.31034910086610453511947107938, 4.00197664824122313983798470526, 4.18193711501116236739734378419, 4.84582754058516930157914867655, 4.97724897268879851068015874740, 5.61619938303031494470719440714, 6.02293828762766177802572178255, 6.20101623033584227621280417147, 6.76349931858681190969506761185, 7.02303209470808462205207979005, 7.52444368932463856971907750757, 7.86296963177424278798038184268, 8.087689005730467510298560007433, 8.483752216984281511831028398115, 8.934058520122654096880560447025
