| L(s) = 1 | + 3-s + 4·5-s + 9-s + 4·15-s − 4·17-s + 2·25-s + 27-s + 12·37-s + 12·41-s + 8·43-s + 4·45-s − 7·49-s − 4·51-s − 8·59-s − 8·67-s + 2·75-s − 16·79-s + 81-s + 8·83-s − 16·85-s + 12·89-s + 36·101-s − 4·109-s + 12·111-s − 6·121-s + 12·123-s − 28·125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.03·15-s − 0.970·17-s + 2/5·25-s + 0.192·27-s + 1.97·37-s + 1.87·41-s + 1.21·43-s + 0.596·45-s − 49-s − 0.560·51-s − 1.04·59-s − 0.977·67-s + 0.230·75-s − 1.80·79-s + 1/9·81-s + 0.878·83-s − 1.73·85-s + 1.27·89-s + 3.58·101-s − 0.383·109-s + 1.13·111-s − 0.545·121-s + 1.08·123-s − 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480486475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480486475\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.541792988687802337799065273854, −9.172969029955001041869475015652, −9.065170023749915638598404079220, −8.228097915824748265448934137029, −7.62403207119791885504595168937, −7.33253412720106144652855023371, −6.30552876175851347232907272447, −6.14969218484336320739052245813, −5.77649722095232052155724795965, −4.83326855247570334999188527992, −4.41794336687255381402125784080, −3.60246495583988551500687864369, −2.51659494204614894722398853037, −2.34965244557392920909834502109, −1.35556028534757571668534423075,
1.35556028534757571668534423075, 2.34965244557392920909834502109, 2.51659494204614894722398853037, 3.60246495583988551500687864369, 4.41794336687255381402125784080, 4.83326855247570334999188527992, 5.77649722095232052155724795965, 6.14969218484336320739052245813, 6.30552876175851347232907272447, 7.33253412720106144652855023371, 7.62403207119791885504595168937, 8.228097915824748265448934137029, 9.065170023749915638598404079220, 9.172969029955001041869475015652, 9.541792988687802337799065273854
