| L(s) = 1 | + 21·3-s − 81·5-s + 98·7-s + 198·9-s − 121·11-s − 824·13-s − 1.70e3·15-s + 978·17-s + 2.14e3·19-s + 2.05e3·21-s + 3.69e3·23-s + 3.43e3·25-s − 945·27-s − 3.48e3·29-s − 7.81e3·31-s − 2.54e3·33-s − 7.93e3·35-s + 1.35e4·37-s − 1.73e4·39-s + 6.49e3·41-s − 1.42e4·43-s − 1.60e4·45-s − 2.03e4·47-s − 7.20e3·49-s + 2.05e4·51-s + 366·53-s + 9.80e3·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 1.44·5-s + 0.755·7-s + 0.814·9-s − 0.301·11-s − 1.35·13-s − 1.95·15-s + 0.820·17-s + 1.35·19-s + 1.01·21-s + 1.45·23-s + 1.09·25-s − 0.249·27-s − 0.768·29-s − 1.46·31-s − 0.406·33-s − 1.09·35-s + 1.63·37-s − 1.82·39-s + 0.603·41-s − 1.17·43-s − 1.18·45-s − 1.34·47-s − 3/7·49-s + 1.10·51-s + 0.0178·53-s + 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 7 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 81 T + p^{5} T^{2} \) |
| 7 | \( 1 - 2 p^{2} T + p^{5} T^{2} \) |
| 13 | \( 1 + 824 T + p^{5} T^{2} \) |
| 17 | \( 1 - 978 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2140 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3699 T + p^{5} T^{2} \) |
| 29 | \( 1 + 120 p T + p^{5} T^{2} \) |
| 31 | \( 1 + 7813 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13597 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6492 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14234 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20352 T + p^{5} T^{2} \) |
| 53 | \( 1 - 366 T + p^{5} T^{2} \) |
| 59 | \( 1 + 9825 T + p^{5} T^{2} \) |
| 61 | \( 1 + 26132 T + p^{5} T^{2} \) |
| 67 | \( 1 + 17093 T + p^{5} T^{2} \) |
| 71 | \( 1 + 23583 T + p^{5} T^{2} \) |
| 73 | \( 1 + 35176 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42490 T + p^{5} T^{2} \) |
| 83 | \( 1 + 22674 T + p^{5} T^{2} \) |
| 89 | \( 1 + 17145 T + p^{5} T^{2} \) |
| 97 | \( 1 + 30727 T + p^{5} T^{2} \) |
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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
−9.155388655379178570260578852674, −8.178446084573017045520020932402, −7.54035132095649526278867486617, −7.34263913639612052207714179667, −5.33881371342782578155713107089, −4.50805359876278769496149288772, −3.40605756182988527864093395127, −2.82489351452063302234759109151, −1.44246761177699182637336225010, 0,
1.44246761177699182637336225010, 2.82489351452063302234759109151, 3.40605756182988527864093395127, 4.50805359876278769496149288772, 5.33881371342782578155713107089, 7.34263913639612052207714179667, 7.54035132095649526278867486617, 8.178446084573017045520020932402, 9.155388655379178570260578852674
