| L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s − 8-s − 0.999·9-s − 1.41·12-s − 4.24·13-s + 16-s + 5.65·17-s + 0.999·18-s + 4.24·19-s − 6·23-s + 1.41·24-s + 4.24·26-s + 5.65·27-s + 6·29-s + 8.48·31-s − 32-s − 5.65·34-s − 0.999·36-s − 6·37-s − 4.24·38-s + 6·39-s − 8.48·41-s − 12·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.816·3-s + 0.5·4-s + 0.577·6-s − 0.353·8-s − 0.333·9-s − 0.408·12-s − 1.17·13-s + 0.250·16-s + 1.37·17-s + 0.235·18-s + 0.973·19-s − 1.25·23-s + 0.288·24-s + 0.832·26-s + 1.08·27-s + 1.11·29-s + 1.52·31-s − 0.176·32-s − 0.970·34-s − 0.166·36-s − 0.986·37-s − 0.688·38-s + 0.960·39-s − 1.32·41-s − 1.82·43-s + 0.884·46-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 | \( 1 + 1.41T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.337446602544081575207299536413, −7.995605997151710897062972608912, −6.95779844084293422933398018527, −6.37499198610427293486799143533, −5.37455954323228912946435291428, −4.94445039296623239893984358188, −3.47906817582929731367555338216, −2.57842518693119346484393496352, −1.23112608141791982397204180843, 0,
1.23112608141791982397204180843, 2.57842518693119346484393496352, 3.47906817582929731367555338216, 4.94445039296623239893984358188, 5.37455954323228912946435291428, 6.37499198610427293486799143533, 6.95779844084293422933398018527, 7.995605997151710897062972608912, 8.337446602544081575207299536413
