| L(s) = 1 | + 9·3-s − 6·5-s + 108·11-s − 692·13-s − 54·15-s + 1.39e3·17-s + 1.01e3·19-s + 1.53e3·23-s + 3.12e3·25-s − 729·27-s − 7.52e3·29-s + 736·31-s + 972·33-s − 2.05e3·37-s − 6.22e3·39-s − 3.10e4·41-s + 2.20e4·43-s − 4.56e3·47-s + 1.25e4·51-s + 7.96e3·53-s − 648·55-s + 9.10e3·57-s + 7.02e3·59-s − 2.68e4·61-s + 4.15e3·65-s − 5.21e4·67-s + 1.38e4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.107·5-s + 0.269·11-s − 1.13·13-s − 0.0619·15-s + 1.17·17-s + 0.643·19-s + 0.605·23-s + 25-s − 0.192·27-s − 1.66·29-s + 0.137·31-s + 0.155·33-s − 0.246·37-s − 0.655·39-s − 2.88·41-s + 1.82·43-s − 0.301·47-s + 0.677·51-s + 0.389·53-s − 0.0288·55-s + 0.371·57-s + 0.262·59-s − 0.924·61-s + 0.121·65-s − 1.41·67-s + 0.349·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9962765964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9962765964\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 3089 T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 108 T - 149387 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 346 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1398 T + 534547 T^{2} - 1398 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 1012 T - 1451955 T^{2} - 1012 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1536 T - 4077047 T^{2} - 1536 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3762 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 736 T - 28087455 T^{2} - 736 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2054 T - 65125041 T^{2} + 2054 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 15534 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11036 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4560 T - 208551407 T^{2} + 4560 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7962 T - 354802049 T^{2} - 7962 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7020 T - 665643899 T^{2} - 7020 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26870 T - 122599401 T^{2} + 26870 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 52148 T + 1369288797 T^{2} + 52148 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2544 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9766 T - 1977696837 T^{2} - 9766 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 68672 T + 1638787185 T^{2} + 68672 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 61668 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 41454 T - 3865625333 T^{2} - 41454 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 111262 T + p^{5} T^{2} )^{2} \) |
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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
−10.30768616017024427981387059803, −9.459658367681639021872237564821, −9.458380857217662344581914987080, −8.700581327075073025105368669581, −8.576406273930600003478011070247, −7.70714624427221395292346851693, −7.63733509769275300760698128395, −6.98877939840209673341032484440, −6.86024127506645067373861994751, −5.84330521272567582879225213185, −5.65413034633560917057799552898, −4.91447959557260422094010899444, −4.74500165619026516235106025757, −3.70386519203619722944042202616, −3.60532491933222595242205452106, −2.71585332982859187762490226605, −2.61783149160094153102135682315, −1.40417024146723616154567069019, −1.38031929112813711367364517213, −0.20817287757675566114708184526,
0.20817287757675566114708184526, 1.38031929112813711367364517213, 1.40417024146723616154567069019, 2.61783149160094153102135682315, 2.71585332982859187762490226605, 3.60532491933222595242205452106, 3.70386519203619722944042202616, 4.74500165619026516235106025757, 4.91447959557260422094010899444, 5.65413034633560917057799552898, 5.84330521272567582879225213185, 6.86024127506645067373861994751, 6.98877939840209673341032484440, 7.63733509769275300760698128395, 7.70714624427221395292346851693, 8.576406273930600003478011070247, 8.700581327075073025105368669581, 9.458380857217662344581914987080, 9.459658367681639021872237564821, 10.30768616017024427981387059803
