| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 66·5-s + 36·6-s + 64·8-s + 81·9-s + 264·10-s − 60·11-s + 144·12-s + 658·13-s + 594·15-s + 256·16-s + 414·17-s + 324·18-s − 956·19-s + 1.05e3·20-s − 240·22-s + 600·23-s + 576·24-s + 1.23e3·25-s + 2.63e3·26-s + 729·27-s + 5.57e3·29-s + 2.37e3·30-s + 3.59e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.18·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.834·10-s − 0.149·11-s + 0.288·12-s + 1.07·13-s + 0.681·15-s + 1/4·16-s + 0.347·17-s + 0.235·18-s − 0.607·19-s + 0.590·20-s − 0.105·22-s + 0.236·23-s + 0.204·24-s + 0.393·25-s + 0.763·26-s + 0.192·27-s + 1.23·29-s + 0.481·30-s + 0.671·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.424389917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.424389917\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 66 T + p^{5} T^{2} \) |
| 11 | \( 1 + 60 T + p^{5} T^{2} \) |
| 13 | \( 1 - 658 T + p^{5} T^{2} \) |
| 17 | \( 1 - 414 T + p^{5} T^{2} \) |
| 19 | \( 1 + 956 T + p^{5} T^{2} \) |
| 23 | \( 1 - 600 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 + 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 + 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 - 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 + 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 - 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 - 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 - 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 + 166082 T + p^{5} T^{2} \) |
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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
−10.79617552596184611305484289769, −10.11630774462742478324994208019, −9.029555867670381142356202516077, −8.112718798013106437475140949894, −6.73027049925005514796468756958, −5.97063122220731253106996152904, −4.86476246428133923610934764365, −3.55961191213915360099516830949, −2.42785062549074626720126549827, −1.32813200601993022889986711952,
1.32813200601993022889986711952, 2.42785062549074626720126549827, 3.55961191213915360099516830949, 4.86476246428133923610934764365, 5.97063122220731253106996152904, 6.73027049925005514796468756958, 8.112718798013106437475140949894, 9.029555867670381142356202516077, 10.11630774462742478324994208019, 10.79617552596184611305484289769
