| L(s) = 1 | + 256·4-s + 239·7-s + 5.64e4·13-s + 6.55e4·16-s + 1.00e5·19-s + 3.90e5·25-s + 6.11e4·28-s − 1.80e6·31-s − 3.46e6·37-s + 3.49e6·43-s − 5.70e6·49-s + 1.44e7·52-s + 2.41e7·61-s + 1.67e7·64-s − 3.18e7·67-s − 5.52e7·73-s + 2.57e7·76-s − 5.60e7·79-s + 1.34e7·91-s − 9.47e7·97-s + 1.00e8·100-s + 1.68e8·103-s + 2.03e8·109-s + 1.56e7·112-s + ⋯ |
| L(s) = 1 | + 4-s + 0.0995·7-s + 1.97·13-s + 16-s + 0.771·19-s + 25-s + 0.0995·28-s − 1.95·31-s − 1.85·37-s + 1.02·43-s − 0.990·49-s + 1.97·52-s + 1.74·61-s + 64-s − 1.58·67-s − 1.94·73-s + 0.771·76-s − 1.43·79-s + 0.196·91-s − 1.07·97-s + 100-s + 1.50·103-s + 1.43·109-s + 0.0995·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.347575844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347575844\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 - 239 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 - 56447 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 - 100559 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 + 1809406 T + p^{8} T^{2} \) |
| 37 | \( 1 + 3468481 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 - 3492194 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 - 24133919 T + p^{8} T^{2} \) |
| 67 | \( 1 + 31874833 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 + 55236481 T + p^{8} T^{2} \) |
| 79 | \( 1 + 56007121 T + p^{8} T^{2} \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 + 94775521 T + p^{8} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.73239903185375892960255291064, −14.36467562839506384696253304793, −12.92032777678837239353552034414, −11.48563204413657505096989140001, −10.59085570473154130582860930113, −8.719896246352272155330860101619, −7.14908257691409053534525031189, −5.76115778173633366558578798284, −3.40674642623065975813479474109, −1.44843040786981824246642172847,
1.44843040786981824246642172847, 3.40674642623065975813479474109, 5.76115778173633366558578798284, 7.14908257691409053534525031189, 8.719896246352272155330860101619, 10.59085570473154130582860930113, 11.48563204413657505096989140001, 12.92032777678837239353552034414, 14.36467562839506384696253304793, 15.73239903185375892960255291064
