L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 4·11-s + 2·13-s − 15-s − 6·17-s − 21-s + 8·23-s + 25-s − 27-s − 10·29-s + 8·31-s + 4·33-s + 35-s − 2·37-s − 2·39-s − 2·41-s + 8·43-s + 45-s − 4·47-s + 49-s + 6·51-s − 10·53-s − 4·55-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 1.21·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.539·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.535804269\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535804269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.926495804164752353146793840079, −7.22182382615052954409275031871, −6.51999660823106657578059342997, −5.86415854518889917156238655889, −5.05001002563478321843210093168, −4.70317531918579253702889563347, −3.61190358530975808146015901230, −2.60752231442130545282685053623, −1.83252286781558621374538897581, −0.64979978751713250743861929056,
0.64979978751713250743861929056, 1.83252286781558621374538897581, 2.60752231442130545282685053623, 3.61190358530975808146015901230, 4.70317531918579253702889563347, 5.05001002563478321843210093168, 5.86415854518889917156238655889, 6.51999660823106657578059342997, 7.22182382615052954409275031871, 7.926495804164752353146793840079