L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3.61·7-s − 8-s + 9-s − 10-s − 12-s − 1.85·13-s − 3.61·14-s − 15-s + 16-s + 2.47·17-s − 18-s − 1.85·19-s + 20-s − 3.61·21-s − 6.38·23-s + 24-s + 25-s + 1.85·26-s − 27-s + 3.61·28-s − 2.76·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s − 0.514·13-s − 0.966·14-s − 0.258·15-s + 0.250·16-s + 0.599·17-s − 0.235·18-s − 0.425·19-s + 0.223·20-s − 0.789·21-s − 1.33·23-s + 0.204·24-s + 0.200·25-s + 0.363·26-s − 0.192·27-s + 0.683·28-s − 0.513·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 3.61T + 7T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 9.85T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 + 8.56T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.168026791023924232864443813523, −7.55024107069850845206151275528, −6.79687078292019141892689253069, −5.91219798273304796756925219005, −5.24381087419757572739366178540, −4.55053508483697630566564583681, −3.39288131452448385928195547603, −1.98947832132654413895585714556, −1.55371866710300917224544928674, 0,
1.55371866710300917224544928674, 1.98947832132654413895585714556, 3.39288131452448385928195547603, 4.55053508483697630566564583681, 5.24381087419757572739366178540, 5.91219798273304796756925219005, 6.79687078292019141892689253069, 7.55024107069850845206151275528, 8.168026791023924232864443813523