L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 3·11-s + 2·12-s − 5·13-s + 16-s − 6·17-s − 18-s − 19-s − 3·22-s − 3·23-s − 2·24-s + 5·26-s − 4·27-s − 6·29-s − 4·31-s − 32-s + 6·33-s + 6·34-s + 36-s − 11·37-s + 38-s − 10·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s − 0.639·22-s − 0.625·23-s − 0.408·24-s + 0.980·26-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.04·33-s + 1.02·34-s + 1/6·36-s − 1.80·37-s + 0.162·38-s − 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.702631180067393503939211308400, −7.907181421919408302592913909804, −7.23269337954175990719972923658, −6.57541833753328061987010774960, −5.48476176291162715657076790272, −4.31421472789520038093756231803, −3.50917816540482049947732806059, −2.39448033478665306770920232105, −1.86657181975118971053975761529, 0,
1.86657181975118971053975761529, 2.39448033478665306770920232105, 3.50917816540482049947732806059, 4.31421472789520038093756231803, 5.48476176291162715657076790272, 6.57541833753328061987010774960, 7.23269337954175990719972923658, 7.907181421919408302592913909804, 8.702631180067393503939211308400