L(s) = 1 | − 2·2-s + 2·4-s + 3·7-s + 2·11-s + 5·13-s − 6·14-s − 4·16-s − 8·17-s + 19-s − 4·22-s + 6·23-s − 10·26-s + 6·28-s − 2·29-s + 8·32-s + 16·34-s − 5·37-s − 2·38-s + 10·41-s − 4·43-s + 4·44-s − 12·46-s + 4·47-s + 2·49-s + 10·52-s − 2·53-s + 4·58-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 1.13·7-s + 0.603·11-s + 1.38·13-s − 1.60·14-s − 16-s − 1.94·17-s + 0.229·19-s − 0.852·22-s + 1.25·23-s − 1.96·26-s + 1.13·28-s − 0.371·29-s + 1.41·32-s + 2.74·34-s − 0.821·37-s − 0.324·38-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.76·46-s + 0.583·47-s + 2/7·49-s + 1.38·52-s − 0.274·53-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8795221474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8795221474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−10.60210395198651996206865981929, −9.311658519936826320807918208179, −8.803059168893277650788213094384, −8.221165697293086925362136863926, −7.18969871091542681332622621187, −6.42086327425032713317527144072, −4.99553531604565819224928500769, −3.96731437635022230303879227393, −2.13719878609626222551121342745, −1.06897770472745757504364106971,
1.06897770472745757504364106971, 2.13719878609626222551121342745, 3.96731437635022230303879227393, 4.99553531604565819224928500769, 6.42086327425032713317527144072, 7.18969871091542681332622621187, 8.221165697293086925362136863926, 8.803059168893277650788213094384, 9.311658519936826320807918208179, 10.60210395198651996206865981929