L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 3·12-s − 2·13-s + 16-s − 2·17-s + 6·18-s − 2·19-s − 23-s − 3·24-s − 2·26-s − 9·27-s − 29-s + 10·31-s + 32-s − 2·34-s + 6·36-s − 8·37-s − 2·38-s + 6·39-s − 3·41-s + 5·43-s − 46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.866·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.458·19-s − 0.208·23-s − 0.612·24-s − 0.392·26-s − 1.73·27-s − 0.185·29-s + 1.79·31-s + 0.176·32-s − 0.342·34-s + 36-s − 1.31·37-s − 0.324·38-s + 0.960·39-s − 0.468·41-s + 0.762·43-s − 0.147·46-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 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.457694153610101181085156869273, −7.37745575263392719142240303901, −6.75520013012546474099032853352, −6.09930638726381028968090665334, −5.43695427028859428637020995663, −4.66130201500720418654712031616, −4.12258943022616777089154137978, −2.71717860339144413618601750928, −1.43058636493935704101488739689, 0,
1.43058636493935704101488739689, 2.71717860339144413618601750928, 4.12258943022616777089154137978, 4.66130201500720418654712031616, 5.43695427028859428637020995663, 6.09930638726381028968090665334, 6.75520013012546474099032853352, 7.37745575263392719142240303901, 8.457694153610101181085156869273