| L(s) = 1 | − 3·3-s + 6·9-s − 2·11-s − 6·13-s + 2·17-s + 9·23-s − 9·27-s + 3·29-s − 2·31-s + 6·33-s − 8·37-s + 18·39-s − 5·41-s − 43-s + 8·47-s − 6·51-s − 4·53-s + 8·59-s − 7·61-s + 3·67-s − 27·69-s + 8·71-s + 14·73-s + 4·79-s + 9·81-s − 83-s − 9·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.603·11-s − 1.66·13-s + 0.485·17-s + 1.87·23-s − 1.73·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s + 2.88·39-s − 0.780·41-s − 0.152·43-s + 1.16·47-s − 0.840·51-s − 0.549·53-s + 1.04·59-s − 0.896·61-s + 0.366·67-s − 3.25·69-s + 0.949·71-s + 1.63·73-s + 0.450·79-s + 81-s − 0.109·83-s − 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−7.54414826077870075855418777440, −7.08323146641366109516368479981, −6.49965521078597254439212082546, −5.44868416696814366316750824605, −5.13470987898625565072110551265, −4.60345544108394122125661053789, −3.37123726431502200535783657929, −2.28476409329928630042721202778, −1.01504080238500114681392768306, 0,
1.01504080238500114681392768306, 2.28476409329928630042721202778, 3.37123726431502200535783657929, 4.60345544108394122125661053789, 5.13470987898625565072110551265, 5.44868416696814366316750824605, 6.49965521078597254439212082546, 7.08323146641366109516368479981, 7.54414826077870075855418777440
