L(s) = 1 | − 27·3-s − 92·7-s + 729·9-s + 3.45e3·11-s − 4.61e3·13-s − 1.75e4·17-s − 1.30e3·19-s + 2.48e3·21-s − 1.40e4·23-s − 1.96e4·27-s + 1.74e5·29-s + 1.89e5·31-s − 9.33e4·33-s − 2.79e5·37-s + 1.24e5·39-s + 3.57e5·41-s + 2.83e5·43-s − 1.01e5·47-s − 8.15e5·49-s + 4.72e5·51-s + 3.92e5·53-s + 3.51e4·57-s − 5.39e5·59-s − 1.94e6·61-s − 6.70e4·63-s + 1.85e6·67-s + 3.80e5·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.101·7-s + 1/3·9-s + 0.782·11-s − 0.581·13-s − 0.864·17-s − 0.0434·19-s + 0.0585·21-s − 0.241·23-s − 0.192·27-s + 1.32·29-s + 1.14·31-s − 0.452·33-s − 0.907·37-s + 0.335·39-s + 0.810·41-s + 0.544·43-s − 0.142·47-s − 0.989·49-s + 0.498·51-s + 0.362·53-s + 0.0251·57-s − 0.342·59-s − 1.09·61-s − 0.0337·63-s + 0.753·67-s + 0.139·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 92 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3456 T + p^{7} T^{2} \) |
| 13 | \( 1 + 4610 T + p^{7} T^{2} \) |
| 17 | \( 1 + 17502 T + p^{7} T^{2} \) |
| 19 | \( 1 + 1300 T + p^{7} T^{2} \) |
| 23 | \( 1 + 14088 T + p^{7} T^{2} \) |
| 29 | \( 1 - 174306 T + p^{7} T^{2} \) |
| 31 | \( 1 - 189824 T + p^{7} T^{2} \) |
| 37 | \( 1 + 279506 T + p^{7} T^{2} \) |
| 41 | \( 1 - 357690 T + p^{7} T^{2} \) |
| 43 | \( 1 - 283852 T + p^{7} T^{2} \) |
| 47 | \( 1 + 101688 T + p^{7} T^{2} \) |
| 53 | \( 1 - 392574 T + p^{7} T^{2} \) |
| 59 | \( 1 + 539904 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1946338 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1855852 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1683840 T + p^{7} T^{2} \) |
| 73 | \( 1 - 56302 p T + p^{7} T^{2} \) |
| 79 | \( 1 + 4565008 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5444868 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5461230 T + p^{7} T^{2} \) |
| 97 | \( 1 - 12074686 T + p^{7} T^{2} \) |
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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
−10.10202124253878115501346541250, −9.210761604785611892760542376939, −8.142066392587397247221472990205, −6.90228642599620819738067287283, −6.24818989572178836100447098360, −4.97525371134955765184338010683, −4.07977028089842895171027456861, −2.61202001326826220908516523805, −1.24008959697347467287569322786, 0,
1.24008959697347467287569322786, 2.61202001326826220908516523805, 4.07977028089842895171027456861, 4.97525371134955765184338010683, 6.24818989572178836100447098360, 6.90228642599620819738067287283, 8.142066392587397247221472990205, 9.210761604785611892760542376939, 10.10202124253878115501346541250