L(s) = 1 | − 3-s + 9-s + 2·11-s + 2·13-s − 4·17-s − 4·19-s + 6·23-s − 5·25-s − 27-s − 2·29-s − 2·33-s − 6·37-s − 2·39-s − 8·41-s + 8·43-s − 4·47-s + 4·51-s − 6·53-s + 4·57-s + 14·61-s − 4·67-s − 6·69-s + 2·71-s + 2·73-s + 5·75-s − 4·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.970·17-s − 0.917·19-s + 1.25·23-s − 25-s − 0.192·27-s − 0.371·29-s − 0.348·33-s − 0.986·37-s − 0.320·39-s − 1.24·41-s + 1.21·43-s − 0.583·47-s + 0.560·51-s − 0.824·53-s + 0.529·57-s + 1.79·61-s − 0.488·67-s − 0.722·69-s + 0.237·71-s + 0.234·73-s + 0.577·75-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.978612217860669975132035765741, −6.91853482796032508358658511458, −6.61385731110797127452616848075, −5.78180731932062267120229177749, −5.00739525122673721225572264562, −4.19258498203741051537016461394, −3.51673300990944375771370036258, −2.27637878640237515977372161758, −1.33794435930813983309121693954, 0,
1.33794435930813983309121693954, 2.27637878640237515977372161758, 3.51673300990944375771370036258, 4.19258498203741051537016461394, 5.00739525122673721225572264562, 5.78180731932062267120229177749, 6.61385731110797127452616848075, 6.91853482796032508358658511458, 7.978612217860669975132035765741