L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s − 6·11-s + 12-s − 4·13-s + 16-s − 2·18-s − 2·19-s − 6·22-s + 3·23-s + 24-s − 4·26-s − 5·27-s − 3·29-s − 8·31-s + 32-s − 6·33-s − 2·36-s + 4·37-s − 2·38-s − 4·39-s − 9·41-s + 7·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 1.27·22-s + 0.625·23-s + 0.204·24-s − 0.784·26-s − 0.962·27-s − 0.557·29-s − 1.43·31-s + 0.176·32-s − 1.04·33-s − 1/3·36-s + 0.657·37-s − 0.324·38-s − 0.640·39-s − 1.40·41-s + 1.06·43-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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−8.493020359626800107127698839968, −7.58057077276815775951499435952, −7.29055981166105119399090794834, −5.97680589119129465764263076664, −5.33827211027875512451088784271, −4.67969109619292585698671062404, −3.51066995760779525212091377589, −2.70599094868407403174498716052, −2.11959998686408688736060734374, 0,
2.11959998686408688736060734374, 2.70599094868407403174498716052, 3.51066995760779525212091377589, 4.67969109619292585698671062404, 5.33827211027875512451088784271, 5.97680589119129465764263076664, 7.29055981166105119399090794834, 7.58057077276815775951499435952, 8.493020359626800107127698839968