| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 1.23·7-s − 8-s + 9-s + 2·11-s + 12-s − 5.23·13-s − 1.23·14-s + 16-s − 17-s − 18-s − 6.47·19-s + 1.23·21-s − 2·22-s − 4·23-s − 24-s + 5.23·26-s + 27-s + 1.23·28-s − 4·29-s + 2.47·31-s − 32-s + 2·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.467·7-s − 0.353·8-s + 0.333·9-s + 0.603·11-s + 0.288·12-s − 1.45·13-s − 0.330·14-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.48·19-s + 0.269·21-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 1.02·26-s + 0.192·27-s + 0.233·28-s − 0.742·29-s + 0.444·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
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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
−8.337485126059074815061240004163, −8.120301109314120931509754123838, −6.96686921318522211159659429652, −6.63098031149746496571831001938, −5.34981077663953021974146322552, −4.47738089346060040263144933860, −3.54270894406654313486798345220, −2.33188837955964504683728384855, −1.73080267571195772374368158473, 0,
1.73080267571195772374368158473, 2.33188837955964504683728384855, 3.54270894406654313486798345220, 4.47738089346060040263144933860, 5.34981077663953021974146322552, 6.63098031149746496571831001938, 6.96686921318522211159659429652, 8.120301109314120931509754123838, 8.337485126059074815061240004163
