| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3.35·7-s + 8-s + 9-s − 10-s + 1.69·11-s + 12-s + 3.35·14-s − 15-s + 16-s + 0.939·17-s + 18-s − 4.85·19-s − 20-s + 3.35·21-s + 1.69·22-s + 4.04·23-s + 24-s + 25-s + 27-s + 3.35·28-s − 8.12·29-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 + 1.26·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.510·11-s + 0.288·12-s + 0.897·14-s − 0.258·15-s + 0.250·16-s + 0.227·17-s + 0.235·18-s − 1.11·19-s − 0.223·20-s + 0.732·21-s + 0.360·22-s + 0.844·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.634·28-s − 1.50·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.547412178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.547412178\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 17 | \( 1 - 0.939T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 4.08T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 - 4.02T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 0.417T + 59T^{2} \) |
| 61 | \( 1 - 0.198T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 5.15T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.042090847558868501753061953150, −7.62903503615413829554150673602, −6.84439824012692477389042504955, −5.99916422967516959369248192442, −5.15176396055753321546959436186, −4.32024046025299993374760019575, −3.99766922502014349560981145629, −2.88220139527315090851176651076, −2.07214080818837533350997329564, −1.09726644194426819937869221615,
1.09726644194426819937869221615, 2.07214080818837533350997329564, 2.88220139527315090851176651076, 3.99766922502014349560981145629, 4.32024046025299993374760019575, 5.15176396055753321546959436186, 5.99916422967516959369248192442, 6.84439824012692477389042504955, 7.62903503615413829554150673602, 8.042090847558868501753061953150
