L(s) = 1 | + 3-s + 4·5-s + 9-s − 6·11-s − 5·13-s + 4·15-s + 2·17-s − 19-s − 6·23-s + 11·25-s + 27-s − 3·31-s − 6·33-s − 3·37-s − 5·39-s − 6·41-s − 5·43-s + 4·45-s − 4·47-s + 2·51-s + 6·53-s − 24·55-s − 57-s + 6·59-s + 2·61-s − 20·65-s − 7·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.80·11-s − 1.38·13-s + 1.03·15-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 0.538·31-s − 1.04·33-s − 0.493·37-s − 0.800·39-s − 0.937·41-s − 0.762·43-s + 0.596·45-s − 0.583·47-s + 0.280·51-s + 0.824·53-s − 3.23·55-s − 0.132·57-s + 0.781·59-s + 0.256·61-s − 2.48·65-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.38675901359923775297323914553, −6.71211902479141432223357047503, −5.88484572578998833010254648823, −5.17033142969911593212871205012, −4.98807755077435581991657763500, −3.67612153602993300128122363301, −2.62388265008600548640007629296, −2.36494144908712249974554568805, −1.59133075911311899636657158204, 0,
1.59133075911311899636657158204, 2.36494144908712249974554568805, 2.62388265008600548640007629296, 3.67612153602993300128122363301, 4.98807755077435581991657763500, 5.17033142969911593212871205012, 5.88484572578998833010254648823, 6.71211902479141432223357047503, 7.38675901359923775297323914553