| L(s) = 1 | + 2-s + 3-s + 4-s − 1.19·5-s + 6-s + 7-s + 8-s + 9-s − 1.19·10-s − 1.55·11-s + 12-s + 14-s − 1.19·15-s + 16-s − 5.74·17-s + 18-s − 4.93·19-s − 1.19·20-s + 21-s − 1.55·22-s − 0.417·23-s + 24-s − 3.56·25-s + 27-s + 28-s − 2.46·29-s − 1.19·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.535·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.378·10-s − 0.468·11-s + 0.288·12-s + 0.267·14-s − 0.309·15-s + 0.250·16-s − 1.39·17-s + 0.235·18-s − 1.13·19-s − 0.267·20-s + 0.218·21-s − 0.331·22-s − 0.0871·23-s + 0.204·24-s − 0.712·25-s + 0.192·27-s + 0.188·28-s − 0.458·29-s − 0.218·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.19T + 5T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 + 0.417T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 + 9.06T + 83T^{2} \) |
| 89 | \( 1 + 3.67T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.59661896356100301442237919312, −6.82377829606075238733671206257, −6.28237378449982462111439412594, −5.32927300650717325661312646451, −4.38491301498735226811210107398, −4.24741445824610656116778087691, −3.15470782148629430912775964496, −2.42477940547994516899024362149, −1.63896154928425624085374196495, 0,
1.63896154928425624085374196495, 2.42477940547994516899024362149, 3.15470782148629430912775964496, 4.24741445824610656116778087691, 4.38491301498735226811210107398, 5.32927300650717325661312646451, 6.28237378449982462111439412594, 6.82377829606075238733671206257, 7.59661896356100301442237919312
