L(s) = 1 | + 3-s − 4·5-s + 9-s + 2·11-s + 13-s − 4·15-s + 2·17-s − 8·19-s + 4·23-s + 11·25-s + 27-s + 6·29-s − 4·31-s + 2·33-s − 6·37-s + 39-s − 12·41-s − 4·43-s − 4·45-s − 6·47-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s − 8·57-s + 14·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 1.03·15-s + 0.485·17-s − 1.83·19-s + 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s − 1.05·57-s + 1.82·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.423418847778347999015510445934, −8.039381956247822929267892267786, −6.90842466906101662199467048261, −6.68867852483491298922981743769, −5.14437425119067985622882248862, −4.30246855345630396897009783280, −3.68129802131091868207255830596, −2.96468384245954072400571713270, −1.50180093220136700598375543301, 0,
1.50180093220136700598375543301, 2.96468384245954072400571713270, 3.68129802131091868207255830596, 4.30246855345630396897009783280, 5.14437425119067985622882248862, 6.68867852483491298922981743769, 6.90842466906101662199467048261, 8.039381956247822929267892267786, 8.423418847778347999015510445934