| L(s) = 1 | − 2·3-s − 5-s − 3·7-s + 9-s + 5·11-s + 4·13-s + 2·15-s − 3·17-s + 6·21-s + 8·23-s − 4·25-s + 4·27-s + 2·29-s − 4·31-s − 10·33-s + 3·35-s − 10·37-s − 8·39-s − 10·41-s + 43-s − 45-s − 47-s + 2·49-s + 6·51-s + 4·53-s − 5·55-s − 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.516·15-s − 0.727·17-s + 1.30·21-s + 1.66·23-s − 4/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s − 1.74·33-s + 0.507·35-s − 1.64·37-s − 1.28·39-s − 1.56·41-s + 0.152·43-s − 0.149·45-s − 0.145·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s − 0.674·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.042469067479731915872544923229, −8.597280713020450652266450131967, −7.06517613032516495660782139166, −6.61740221810831101558532213495, −6.00860191803297233312976968768, −5.02061333559614135342377447774, −3.92465502465422783797773427320, −3.23414224716771269543260300521, −1.36067895800170659578793446698, 0,
1.36067895800170659578793446698, 3.23414224716771269543260300521, 3.92465502465422783797773427320, 5.02061333559614135342377447774, 6.00860191803297233312976968768, 6.61740221810831101558532213495, 7.06517613032516495660782139166, 8.597280713020450652266450131967, 9.042469067479731915872544923229
