| L(s) = 1 | − 3·9-s − 2·11-s + 2·13-s − 2·19-s + 8·23-s + 29-s + 2·31-s + 4·37-s − 10·41-s − 4·43-s − 12·47-s − 7·49-s + 6·53-s − 12·59-s − 10·61-s − 12·67-s + 12·71-s − 12·73-s + 2·79-s + 9·81-s + 4·83-s − 10·89-s − 8·97-s + 6·99-s − 6·101-s − 4·107-s + 6·109-s + ⋯ |
| L(s) = 1 | − 9-s − 0.603·11-s + 0.554·13-s − 0.458·19-s + 1.66·23-s + 0.185·29-s + 0.359·31-s + 0.657·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s − 1.46·67-s + 1.42·71-s − 1.40·73-s + 0.225·79-s + 81-s + 0.439·83-s − 1.05·89-s − 0.812·97-s + 0.603·99-s − 0.597·101-s − 0.386·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.426266561548849263818126665219, −7.78747203827088311187613738607, −6.76112273763255746868599958761, −6.15452921521928361854394263073, −5.23860109683373417828298227637, −4.62890862797498589560907518225, −3.33011214354228204499572453631, −2.80491934880049729005942168277, −1.49565702759520034440517571350, 0,
1.49565702759520034440517571350, 2.80491934880049729005942168277, 3.33011214354228204499572453631, 4.62890862797498589560907518225, 5.23860109683373417828298227637, 6.15452921521928361854394263073, 6.76112273763255746868599958761, 7.78747203827088311187613738607, 8.426266561548849263818126665219
