L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 2·8-s + 9-s − 2·11-s − 4·14-s − 4·16-s + 2·18-s − 4·22-s − 4·23-s + 25-s − 2·28-s − 2·29-s − 2·32-s + 36-s + 2·43-s − 2·44-s − 8·46-s + 3·49-s + 2·50-s − 2·53-s + 4·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + ⋯ |
L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 2·8-s + 9-s − 2·11-s − 4·14-s − 4·16-s + 2·18-s − 4·22-s − 4·23-s + 25-s − 2·28-s − 2·29-s − 2·32-s + 36-s + 2·43-s − 2·44-s − 8·46-s + 3·49-s + 2·50-s − 2·53-s + 4·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3874767902\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3874767902\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.950442625964970499725881326458, −8.782478952760063691003021097310, −8.136743777394137608595560109521, −7.75178268885156786753077014015, −7.38665007018077791666216631276, −7.11746300116069193607360004874, −6.26411059276351546498590140271, −6.21834174073620449732068242903, −5.84251601188439348149896939969, −5.77396384393688512860474928107, −5.03199486010886434058841200530, −4.85533111177461812743520028068, −4.22075374901353123559614346399, −3.84137852850043809542149813079, −3.83820096694568181422187927842, −3.23507497369810900200048463229, −2.60974997414010456890227051542, −2.58095675157051222116791334213, −1.81048874395588961463311634039, −0.24613183546964114335783284909,
0.24613183546964114335783284909, 1.81048874395588961463311634039, 2.58095675157051222116791334213, 2.60974997414010456890227051542, 3.23507497369810900200048463229, 3.83820096694568181422187927842, 3.84137852850043809542149813079, 4.22075374901353123559614346399, 4.85533111177461812743520028068, 5.03199486010886434058841200530, 5.77396384393688512860474928107, 5.84251601188439348149896939969, 6.21834174073620449732068242903, 6.26411059276351546498590140271, 7.11746300116069193607360004874, 7.38665007018077791666216631276, 7.75178268885156786753077014015, 8.136743777394137608595560109521, 8.782478952760063691003021097310, 8.950442625964970499725881326458