| L(s) = 1 | + 3·3-s + 6·5-s + 6·9-s + 18·15-s + 6·17-s + 17·25-s + 9·27-s + 14·37-s + 12·41-s − 8·43-s + 36·45-s − 6·47-s + 18·51-s + 6·59-s − 10·67-s + 51·75-s + 2·79-s + 9·81-s + 24·83-s + 36·85-s − 18·89-s − 18·101-s − 34·109-s + 42·111-s − 5·121-s + 36·123-s + 18·125-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.68·5-s + 2·9-s + 4.64·15-s + 1.45·17-s + 17/5·25-s + 1.73·27-s + 2.30·37-s + 1.87·41-s − 1.21·43-s + 5.36·45-s − 0.875·47-s + 2.52·51-s + 0.781·59-s − 1.22·67-s + 5.88·75-s + 0.225·79-s + 81-s + 2.63·83-s + 3.90·85-s − 1.90·89-s − 1.79·101-s − 3.25·109-s + 3.98·111-s − 0.454·121-s + 3.24·123-s + 1.60·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.12091755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.12091755\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.242280570944438833390391985369, −9.014927468599841897781412369793, −8.412039827013958710060244888459, −8.086279505440205535875573741466, −7.71312140618772068338628695294, −7.45937481871917530508733249701, −6.72779678658119428420783825990, −6.51695994129523155309253348230, −5.94420801722824573037775505314, −5.83982910342749951332326991368, −5.14547305714711686285047590157, −5.04794842379288004639334006590, −4.05061415535510979311250814428, −4.04090923416687042606299194485, −3.08926821699092716004132145370, −2.88235880351380532959796681910, −2.39216350377175868336167143012, −2.08303029070695416526004897607, −1.34806221441531617437249486176, −1.19699530402953887300566004150,
1.19699530402953887300566004150, 1.34806221441531617437249486176, 2.08303029070695416526004897607, 2.39216350377175868336167143012, 2.88235880351380532959796681910, 3.08926821699092716004132145370, 4.04090923416687042606299194485, 4.05061415535510979311250814428, 5.04794842379288004639334006590, 5.14547305714711686285047590157, 5.83982910342749951332326991368, 5.94420801722824573037775505314, 6.51695994129523155309253348230, 6.72779678658119428420783825990, 7.45937481871917530508733249701, 7.71312140618772068338628695294, 8.086279505440205535875573741466, 8.412039827013958710060244888459, 9.014927468599841897781412369793, 9.242280570944438833390391985369
