| L(s) = 1 | + 1.02e3·4-s − 1.18e5·13-s + 7.86e5·16-s + 3.90e6·25-s − 3.31e7·43-s + 7.75e7·49-s − 1.21e8·52-s + 2.35e8·61-s + 5.36e8·64-s + 1.23e9·79-s + 4.00e9·100-s + 1.24e8·103-s + 4.71e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.40e9·169-s − 3.39e10·172-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.14·13-s + 3·16-s + 2·25-s − 1.47·43-s + 1.92·49-s − 2.29·52-s + 2.18·61-s + 4·64-s + 3.56·79-s + 4·100-s + 0.109·103-s + 2·121-s + 0.321·169-s − 2.95·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.446950265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.446950265\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 118370 T + p^{9} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 12580 T + p^{9} T^{2} )( 1 + 12580 T + p^{9} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 976696 T + p^{9} T^{2} )( 1 + 976696 T + p^{9} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1691228 T + p^{9} T^{2} )( 1 + 1691228 T + p^{9} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 15384490 T + p^{9} T^{2} )( 1 + 15384490 T + p^{9} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 16577080 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 117903058 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 112542320 T + p^{9} T^{2} )( 1 + 112542320 T + p^{9} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 296368310 T + p^{9} T^{2} )( 1 + 296368310 T + p^{9} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 616732324 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1288928270 T + p^{9} T^{2} )( 1 + 1288928270 T + p^{9} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93382191262782867459984979219, −11.57438640904300532798689859807, −10.86950954382854481401511468432, −10.67485354113417400549743797361, −10.03506143245476392109264371515, −9.613961966350409916136248839616, −8.659903923434182797061358891346, −8.221971020227304238805651277047, −7.36340770744822227145904160391, −7.19912802304553050352199573067, −6.61636965215688867507044031731, −6.14560115792473967132048747200, −5.28045709406360243641527702761, −4.92626149222058920277591807974, −3.74088979371958876308052522564, −3.15915302384159762811963127441, −2.39706777769090748754365010342, −2.19254435024587181999383837281, −1.21451861381375616016733028640, −0.64518348338399081562300672874,
0.64518348338399081562300672874, 1.21451861381375616016733028640, 2.19254435024587181999383837281, 2.39706777769090748754365010342, 3.15915302384159762811963127441, 3.74088979371958876308052522564, 4.92626149222058920277591807974, 5.28045709406360243641527702761, 6.14560115792473967132048747200, 6.61636965215688867507044031731, 7.19912802304553050352199573067, 7.36340770744822227145904160391, 8.221971020227304238805651277047, 8.659903923434182797061358891346, 9.613961966350409916136248839616, 10.03506143245476392109264371515, 10.67485354113417400549743797361, 10.86950954382854481401511468432, 11.57438640904300532798689859807, 11.93382191262782867459984979219
