| L(s) = 1 | + 2·3-s − 3·7-s + 9-s + 5·11-s − 5·13-s − 4·17-s + 19-s − 6·21-s − 23-s − 4·27-s + 9·29-s − 2·31-s + 10·33-s − 2·37-s − 10·39-s + 3·41-s − 7·43-s − 12·47-s + 2·49-s − 8·51-s + 12·53-s + 2·57-s − 6·59-s − 10·61-s − 3·63-s − 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 1.38·13-s − 0.970·17-s + 0.229·19-s − 1.30·21-s − 0.208·23-s − 0.769·27-s + 1.67·29-s − 0.359·31-s + 1.74·33-s − 0.328·37-s − 1.60·39-s + 0.468·41-s − 1.06·43-s − 1.75·47-s + 2/7·49-s − 1.12·51-s + 1.64·53-s + 0.264·57-s − 0.781·59-s − 1.28·61-s − 0.377·63-s − 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.133167489905966018865209731441, −7.08458420364805452776862694925, −6.75025560154426304611390206945, −5.94841026319287427664945572303, −4.76068229814268971566425078519, −4.03783849567142340119209459538, −3.18297579624874854812101142836, −2.64753609177892676628234600908, −1.61739068946046046710947299703, 0,
1.61739068946046046710947299703, 2.64753609177892676628234600908, 3.18297579624874854812101142836, 4.03783849567142340119209459538, 4.76068229814268971566425078519, 5.94841026319287427664945572303, 6.75025560154426304611390206945, 7.08458420364805452776862694925, 8.133167489905966018865209731441
