| L(s) = 1 | − 3-s + 9-s + 2·13-s − 6·17-s + 4·19-s − 8·23-s − 27-s + 2·29-s + 4·31-s + 10·37-s − 2·39-s + 2·41-s − 4·43-s − 8·47-s − 7·49-s + 6·51-s − 2·53-s − 4·57-s − 8·59-s + 2·61-s − 12·67-s + 8·69-s + 8·71-s + 14·73-s − 12·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.840·51-s − 0.274·53-s − 0.529·57-s − 1.04·59-s + 0.256·61-s − 1.46·67-s + 0.963·69-s + 0.949·71-s + 1.63·73-s − 1.35·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.988978183403189308202946756873, −7.11429101967404679710306373861, −6.26981180748617790233113116456, −5.98584821905842504520139258215, −4.84631664932105969766897928377, −4.35033790248089879503765323739, −3.38892885964447337712213070381, −2.33519869008017404609818198079, −1.30074233558505557059691992467, 0,
1.30074233558505557059691992467, 2.33519869008017404609818198079, 3.38892885964447337712213070381, 4.35033790248089879503765323739, 4.84631664932105969766897928377, 5.98584821905842504520139258215, 6.26981180748617790233113116456, 7.11429101967404679710306373861, 7.988978183403189308202946756873
