| L(s) = 1 | + 0.395·2-s − 3-s − 1.84·4-s − 5-s − 0.395·6-s − 4.18·7-s − 1.51·8-s + 9-s − 0.395·10-s − 4.53·11-s + 1.84·12-s + 0.790·13-s − 1.65·14-s + 15-s + 3.08·16-s − 0.109·17-s + 0.395·18-s + 2.45·19-s + 1.84·20-s + 4.18·21-s − 1.79·22-s + 1.51·24-s + 25-s + 0.312·26-s − 27-s + 7.71·28-s + 0.297·29-s + ⋯ |
| L(s) = 1 | + 0.279·2-s − 0.577·3-s − 0.921·4-s − 0.447·5-s − 0.161·6-s − 1.58·7-s − 0.537·8-s + 0.333·9-s − 0.124·10-s − 1.36·11-s + 0.532·12-s + 0.219·13-s − 0.441·14-s + 0.258·15-s + 0.771·16-s − 0.0264·17-s + 0.0931·18-s + 0.562·19-s + 0.412·20-s + 0.912·21-s − 0.381·22-s + 0.310·24-s + 0.200·25-s + 0.0612·26-s − 0.192·27-s + 1.45·28-s + 0.0552·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.395T + 2T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 0.790T + 13T^{2} \) |
| 17 | \( 1 + 0.109T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 29 | \( 1 - 0.297T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 3.16T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.938T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
| 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
−7.41309146818083323563165505719, −6.66283340648445467490268563971, −6.09914599951319060533317787094, −5.14326675410132171789994381520, −4.98078039806277593249921732613, −3.72404373812526046783580402486, −3.42230537817878378933583190977, −2.48359887947410332915243865833, −0.78292671526058980933498600320, 0,
0.78292671526058980933498600320, 2.48359887947410332915243865833, 3.42230537817878378933583190977, 3.72404373812526046783580402486, 4.98078039806277593249921732613, 5.14326675410132171789994381520, 6.09914599951319060533317787094, 6.66283340648445467490268563971, 7.41309146818083323563165505719
