| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 93·5-s − 36·6-s − 49·7-s + 64·8-s + 81·9-s + 372·10-s − 302·11-s − 144·12-s − 169·13-s − 196·14-s − 837·15-s + 256·16-s − 488·17-s + 324·18-s + 2.05e3·19-s + 1.48e3·20-s + 441·21-s − 1.20e3·22-s + 59·23-s − 576·24-s + 5.52e3·25-s − 676·26-s − 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.66·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.17·10-s − 0.752·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.960·15-s + 1/4·16-s − 0.409·17-s + 0.235·18-s + 1.30·19-s + 0.831·20-s + 0.218·21-s − 0.532·22-s + 0.0232·23-s − 0.204·24-s + 1.76·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.915360644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.915360644\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 93 T + p^{5} T^{2} \) |
| 11 | \( 1 + 302 T + p^{5} T^{2} \) |
| 17 | \( 1 + 488 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2053 T + p^{5} T^{2} \) |
| 23 | \( 1 - 59 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5871 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3861 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12388 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2602 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14221 T + p^{5} T^{2} \) |
| 47 | \( 1 + 21645 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7781 T + p^{5} T^{2} \) |
| 59 | \( 1 - 19072 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13954 T + p^{5} T^{2} \) |
| 67 | \( 1 - 2694 T + p^{5} T^{2} \) |
| 71 | \( 1 - 82032 T + p^{5} T^{2} \) |
| 73 | \( 1 - 6503 T + p^{5} T^{2} \) |
| 79 | \( 1 - 28535 T + p^{5} T^{2} \) |
| 83 | \( 1 - 15019 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41979 T + p^{5} T^{2} \) |
| 97 | \( 1 + 57405 T + p^{5} T^{2} \) |
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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
−9.944785592925614786115180717441, −9.612688043862781438649632375354, −8.134987767750928694307930019746, −6.84334186655392320241116812926, −6.22550362869951504678123493488, −5.36230832545086776772802267502, −4.74977725312544173653801633961, −3.08019766612721338552325304671, −2.18849388548179538541415294718, −0.927872047644358692894540287180,
0.927872047644358692894540287180, 2.18849388548179538541415294718, 3.08019766612721338552325304671, 4.74977725312544173653801633961, 5.36230832545086776772802267502, 6.22550362869951504678123493488, 6.84334186655392320241116812926, 8.134987767750928694307930019746, 9.612688043862781438649632375354, 9.944785592925614786115180717441
